In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
between the three sides of a
right triangle
A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right an ...
. It states that the area of the square whose side is the
hypotenuse
In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equa ...
(the side opposite the
right angle
In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...
) is equal to the sum of the areas of the squares on the other two sides. This
theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...
can be written as an
equation
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
relating the lengths of the sides ''a'', ''b'' and the hypotenuse ''c'', often called the Pythagorean equation:
:
The theorem is named for the
Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...
philosopher
Pythagoras
Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samos, Samian, or simply ; in Ionian Greek; ) was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philosopher and the eponymou ...
, born around 570 BC. The theorem has been
proven
Proven is a rural village in the Belgian province of West Flanders, and a "deelgemeente" of the municipality Poperinge. The village has about 1400 inhabitants.
The church and parish of Proven are named after Saint Victor. The Saint Victor Chur ...
numerous times by many different methods – possibly the most for any mathematical theorem. The proofs are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years.
When
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
is represented by a
Cartesian coordinate system
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
in
analytic geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
Analytic geometry is used in physics and engineerin ...
,
Euclidean distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points.
It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefor ...
satisfies the Pythagorean relation: the squared distance between two points equals the sum of squares of the difference in each coordinate between the points.
The theorem can be
generalized in various ways: to
higher-dimensional space
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordina ...
s, to
spaces that are not Euclidean, to objects that are not right triangles, and to objects that are not triangles at all but
''n''-dimensional solids. The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical
abstruseness
In philosophy, the terms obscurantism and obscurationism describe the anti-intellectual practices of deliberately presenting information in an abstruse and imprecise manner that limits further inquiry and understanding of a subject. There are two ...
,
mystique, or intellectual power; popular references in literature, plays, musicals, songs, stamps, and cartoons abound.
Other forms of the theorem
If ''c'' denotes the
length
Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...
of the hypotenuse and ''a'' and ''b'' denote the two lengths of the legs of a right triangle, then the Pythagorean theorem can be expressed as the Pythagorean equation:
:
If only the lengths of the legs of the right triangle are known but not the hypotenuse, then the length of the hypotenuse can be calculated with the equation
:
If the length of the hypotenuse and of one leg is known, then the length of the other leg can be calculated as
:
or
:
A generalization of this theorem is the
law of cosines
In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states ...
, which allows the
computation
Computation is any type of arithmetic or non-arithmetic calculation that follows a well-defined model (e.g., an algorithm).
Mechanical or electronic devices (or, historically, people) that perform computations are known as ''computers''. An es ...
of the length of any side of any triangle, given the lengths of the other two sides and the angle between them. If the angle between the other sides is a right angle, the law of cosines reduces to the Pythagorean equation.
Proofs using constructed squares
Rearrangement proofs
In one rearrangement proof, two squares are used whose sides have a measure of
and which contain four right triangles whose sides are ''a'', ''b'' and ''c'', with the hypotenuse being ''c''. In the square on the right side, the triangles are placed such that the corners of the square correspond to the corners of the right angle in the triangles, forming a square in the center whose sides are length ''c''. Each outer square has an area of
as well as
, with
representing the total area of the four triangles. Within the big square on the left side, the four triangles are moved to form two similar rectangles with sides of length ''a'' and ''b''. These rectangles in their new position have now delineated two new squares, one having side length ''a'' is formed in the bottom-left corner, and another square of side length ''b'' formed in the top-right corner. In this new position, this left side now has a square of area
as well as
. Since both squares have the area of
it follows that the other measure of the square area also equal each other such that
=
. With the area of the four triangles removed from both side of the equation what remains is
In another proof rectangles in the second box can also be placed such that both have one corner that correspond to consecutive corners of the square. In this way they also form two boxes, this time in consecutive corners, with areas
and
which will again lead to a second square of with the area
.
English mathematician
Sir Thomas Heath gives this proof in his commentary on Proposition I.47 in
Euclid's ''
Elements'', and mentions the proposals of German mathematicians
Carl Anton Bretschneider
Carl Anton Bretschneider (27 May 1808 – 6 November 1878) was a mathematician from Gotha, Germany. Bretschneider worked in geometry, number theory, and history of geometry. He also worked on logarithmic integrals and mathematical tables. He was ...
and
Hermann Hankel
Hermann Hankel (14 February 1839 – 29 August 1873) was a German mathematician. Having worked on mathematical analysis during his career, he is best known for introducing the Hankel transform and the Hankel matrix.
Biography
Hankel was born on 1 ...
that Pythagoras may have known this proof. Heath himself favors a different proposal for a Pythagorean proof, but acknowledges from the outset of his discussion "that the Greek literature which we possess belonging to the first five centuries after Pythagoras contains no statement specifying this or any other particular great geometric discovery to him." Recent scholarship has cast increasing doubt on any sort of role for Pythagoras as a creator of mathematics, although debate about this continues.
Algebraic proofs
The theorem can be proved algebraically using four copies of the same triangle arranged symmetrically around a square with side ''c'', as shown in the lower part of the diagram. This results in a larger square, with side and area . The four triangles and the square side ''c'' must have the same area as the larger square,
:
giving
:
A similar proof uses four copies of a right triangle with sides ''a'', ''b'' and ''c'', arranged inside a square with side ''c'' as in the top half of the diagram.
[
] The triangles are similar with area
, while the small square has side and area . The area of the large square is therefore
:
But this is a square with side ''c'' and area ''c''
2, so
:
:
Other proofs of the theorem
This theorem may have more known proofs than any other (the
law
Law is a set of rules that are created and are enforceable by social or governmental institutions to regulate behavior,Robertson, ''Crimes against humanity'', 90. with its precise definition a matter of longstanding debate. It has been vario ...
of
quadratic reciprocity
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard st ...
being another contender for that distinction); the book ''The Pythagorean Proposition'' contains 370 proofs.
Proof using similar triangles
![Pythagoras similar triangles simplified](https://upload.wikimedia.org/wikipedia/commons/f/fb/Pythagoras_similar_triangles_simplified.svg)
This proof is based on the
proportionality of the sides of three
similar triangles, that is, upon the fact that the
ratio
In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
of any two corresponding sides of similar triangles is the same regardless of the size of the triangles.
Let ''ABC'' represent a right triangle, with the
right angle
In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...
located at ''C'', as shown on the figure. Draw the
altitude
Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
from point ''C'', and call ''H'' its intersection with the side ''AB''. Point ''H'' divides the length of the hypotenuse ''c'' into parts ''d'' and ''e''. The new triangle, ''ACH,'' is
similar to triangle ''ABC'', because they both have a right angle (by definition of the altitude), and they share the angle at ''A'', meaning that the third angle will be the same in both triangles as well, marked as ''θ'' in the figure. By a similar reasoning, the triangle ''CBH'' is also similar to ''ABC''. The proof of similarity of the triangles requires the
triangle postulate
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non-collinear ...
: The sum of the angles in a triangle is two right angles, and is equivalent to the
parallel postulate
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry:
''If a line segment ...
. Similarity of the triangles leads to the equality of ratios of corresponding sides:
:
The first result equates the
cosines of the angles ''θ'', whereas the second result equates their
sine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...
s.
These ratios can be written as
:
Summing these two equalities results in
:
which, after simplification, demonstrates the Pythagorean theorem:
:
The role of this proof in history is the subject of much speculation. The underlying question is why Euclid did not use this proof, but invented another. One
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...
is that the proof by similar triangles involved a theory of proportions, a topic not discussed until later in the ''Elements'', and that the theory of proportions needed further development at that time.
[
This proof first appeared after a computer program was set to check Euclidean proofs.]
Euclid's proof
![Illustration to Euclid's proof of the Pythagorean theorem](https://upload.wikimedia.org/wikipedia/commons/2/26/Illustration_to_Euclid%27s_proof_of_the_Pythagorean_theorem.svg)
In outline, here is how the proof in
Euclid
Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
's ''
Elements'' proceeds. The large square is divided into a left and right rectangle. A triangle is constructed that has half the area of the left rectangle. Then another triangle is constructed that has half the area of the square on the left-most side. These two triangles are shown to be
congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
, proving this square has the same area as the left rectangle. This argument is followed by a similar version for the right rectangle and the remaining square. Putting the two rectangles together to reform the square on the hypotenuse, its area is the same as the sum of the area of the other two squares. The details follow.
Let ''A'', ''B'', ''C'' be the
vertices of a right triangle, with a right angle at ''A''. Drop a perpendicular from ''A'' to the side opposite the hypotenuse in the square on the hypotenuse. That line divides the square on the hypotenuse into two rectangles, each having the same area as one of the two squares on the legs.
For the formal proof, we require four elementary
lemmata:
# If two triangles have two sides of the one equal to two sides of the other, each to each, and the angles included by those sides equal, then the triangles are congruent (
side-angle-side).
# The area of a triangle is half the area of any parallelogram on the same base and having the same altitude.
# The area of a rectangle is equal to the product of two adjacent sides.
# The area of a square is equal to the product of two of its sides (follows from 3).
Next, each top square is related to a triangle congruent with another triangle related in turn to one of two rectangles making up the lower square.
![Illustration to Euclid's proof of the Pythagorean theorem3](https://upload.wikimedia.org/wikipedia/commons/f/f6/Illustration_to_Euclid%27s_proof_of_the_Pythagorean_theorem3.svg)
The proof is as follows:
#Let ACB be a right-angled triangle with right angle CAB.
#On each of the sides BC, AB, and CA, squares are drawn, CBDE, BAGF, and ACIH, in that order. The construction of squares requires the immediately preceding theorems in Euclid, and depends upon the parallel postulate.
[
]
#From A, draw a line parallel to BD and CE. It will perpendicularly intersect BC and DE at K and L, respectively.
#Join CF and AD, to form the triangles BCF and BDA.
#Angles CAB and BAG are both right angles; therefore C, A, and G are
collinear
In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned ...
.
#Angles CBD and FBA are both right angles; therefore angle ABD equals angle FBC, since both are the sum of a right angle and angle ABC.
#Since AB is equal to FB, BD is equal to BC and angle ABD equals angle FBC, triangle ABD must be congruent to triangle FBC.
#Since A-K-L is a straight line, parallel to BD, then rectangle BDLK has twice the area of triangle ABD because they share the base BD and have the same altitude BK, i.e., a line normal to their common base, connecting the parallel lines BD and AL. (lemma 2)
#Since C is collinear with A and G, and this line is parallel to FB, then square BAGF must be twice in area to triangle FBC.
#Therefore, rectangle BDLK must have the same area as square BAGF = AB
2.
#By applying steps 3 to 10 to the other side of the figure, it can be similarly shown that rectangle CKLE must have the same area as square ACIH = AC
2.
#Adding these two results, AB
2 + AC
2 = BD × BK + KL × KC
#Since BD = KL, BD × BK + KL × KC = BD(BK + KC) = BD × BC
#Therefore, AB
2 + AC
2 = BC
2, since CBDE is a square.
This proof, which appears in Euclid's ''Elements'' as that of Proposition 47 in Book 1, demonstrates that the area of the square on the hypotenuse is the sum of the areas of the other two squares.
This is quite distinct from the proof by similarity of triangles, which is conjectured to be the proof that Pythagoras used.
[The proof by Pythagoras probably was not a general one, as the theory of proportions was developed only two centuries after Pythagoras; see ]
Proofs by dissection and rearrangement
Another by rearrangement is given by the middle animation. A large square is formed with area ''c''2, from four identical right triangles with sides ''a'', ''b'' and ''c'', fitted around a small central square. Then two rectangles are formed with sides ''a'' and ''b'' by moving the triangles. Combining the smaller square with these rectangles produces two squares of areas ''a''2 and ''b''2, which must have the same area as the initial large square.
The third, rightmost image also gives a proof. The upper two squares are divided as shown by the blue and green shading, into pieces that when rearranged can be made to fit in the lower square on the hypotenuse – or conversely the large square can be divided as shown into pieces that fill the other two. This way of cutting one figure into pieces and rearranging them to get another figure is called dissection
Dissection (from Latin ' "to cut to pieces"; also called anatomization) is the dismembering of the body of a deceased animal or plant to study its anatomical structure. Autopsy is used in pathology and forensic medicine to determine the cause o ...
. This shows the area of the large square equals that of the two smaller ones.
Einstein's proof by dissection without rearrangement
Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
gave a proof by dissection in which the pieces do not need to be moved. Instead of using a square on the hypotenuse and two squares on the legs, one can use any other shape that includes the hypotenuse, and two similar shapes that each include one of two legs instead of the hypotenuse (see Similar figures on the three sides). In Einstein's proof, the shape that includes the hypotenuse is the right triangle itself. The dissection consists of dropping a perpendicular from the vertex of the right angle of the triangle to the hypotenuse, thus splitting the whole triangle into two parts. Those two parts have the same shape as the original right triangle, and have the legs of the original triangle as their hypotenuses, and the sum of their areas is that of the original triangle. Because the ratio of the area of a right triangle to the square of its hypotenuse is the same for similar triangles, the relationship between the areas of the three triangles holds for the squares of the sides of the large triangle as well.
Proof by area-preserving shearing
As shown in the accompanying animation, area-preserving shear mappings and translations can transform the squares on the sides adjacent to the right-angle onto the square on the hypotenuse, together covering it exactly. Each shear leaves the base and height unchanged, thus leaving the area unchanged too. The translations also leave the area unchanged, as they do not alter the shapes at all. Each square is first sheared into a parallelogram, and then into a rectangle which can be translated onto one section of the square on the hypotenuse.
Algebraic proofs
A related proof was published by future U.S. President James A. Garfield
James Abram Garfield (November 19, 1831 – September 19, 1881) was the 20th president of the United States, serving from March 4, 1881 until his death six months latertwo months after he was shot by an assassin. A lawyer and Civil War gene ...
(then a U.S. Representative
The United States House of Representatives, often referred to as the House of Representatives, the U.S. House, or simply the House, is the lower chamber of the United States Congress, with the Senate being the upper chamber. Together they c ...
) (see diagram).[
Published in a weekly mathematics column: as noted in and i]
A calendar of mathematical dates: April 1, 1876
by V. Frederick Rickey
Instead of a square it uses a trapezoid
A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium ().
A trapezoid is necessarily a Convex polygon, convex quadri ...
, which can be constructed from the square in the second of the above proofs by bisecting along a diagonal of the inner square, to give the trapezoid as shown in the diagram. The area of the trapezoid can be calculated to be half the area of the square, that is
:
The inner square is similarly halved, and there are only two triangles so the proof proceeds as above except for a factor of , which is removed by multiplying by two to give the result.
Proof using differentials
One can arrive at the Pythagorean theorem by studying how changes in a side produce a change in the hypotenuse and employing calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
.[
][
]
The triangle ''ABC'' is a right triangle, as shown in the upper part of the diagram, with ''BC'' the hypotenuse. At the same time the triangle lengths are measured as shown, with the hypotenuse of length ''y'', the side ''AC'' of length ''x'' and the side ''AB'' of length ''a'', as seen in the lower diagram part.
If ''x'' is increased by a small amount ''dx'' by extending the side ''AC'' slightly to ''D'', then ''y'' also increases by ''dy''. These form two sides of a triangle, ''CDE'', which (with ''E'' chosen so ''CE'' is perpendicular to the hypotenuse) is a right triangle approximately similar to ''ABC''. Therefore, the ratios of their sides must be the same, that is:
:
This can be rewritten as , which is a differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
that can be solved by direct integration:
:
giving
:
The constant can be deduced from ''x'' = 0, ''y'' = ''a'' to give the equation
:
This is more of an intuitive proof than a formal one: it can be made more rigorous if proper limits are used in place of ''dx'' and ''dy''.
Converse
The converse
Converse may refer to:
Mathematics and logic
* Converse (logic), the result of reversing the two parts of a definite or implicational statement
** Converse implication, the converse of a material implication
** Converse nonimplication, a logical c ...
of the theorem is also true:[
]
Given a triangle with sides of length ''a'', ''b'', and ''c'', if then the angle between sides ''a'' and ''b'' is a right angle
In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...
.
For any three positive real numbers
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
''a'', ''b'', and ''c'' such that , there exists a triangle with sides ''a'', ''b'' and ''c'' as a consequence of the converse of the triangle inequality.
This converse appears in Euclid's ''Elements'' (Book I, Proposition 48): "If in a triangle the square on one of the sides equals the sum of the squares on the remaining two sides of the triangle, then the angle contained by the remaining two sides of the triangle is right."
It can be proven using the law of cosines
In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states ...
or as follows:
Let ''ABC'' be a triangle with side lengths ''a'', ''b'', and ''c'', with Construct a second triangle with sides of length ''a'' and ''b'' containing a right angle. By the Pythagorean theorem, it follows that the hypotenuse of this triangle has length ''c'' = , the same as the hypotenuse of the first triangle. Since both triangles' sides are the same lengths ''a'', ''b'' and ''c'', the triangles are congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
and must have the same angles. Therefore, the angle between the side of lengths ''a'' and ''b'' in the original triangle is a right angle.
The above proof of the converse makes use of the Pythagorean theorem itself. The converse can also be proven without assuming the Pythagorean theorem.
A corollary
In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another ...
of the Pythagorean theorem's converse is a simple means of determining whether a triangle is right, obtuse, or acute, as follows. Let ''c'' be chosen to be the longest of the three sides and (otherwise there is no triangle according to the triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of degenerate triangles, but ...
). The following statements apply:[
]
* If then the triangle is right.
* If then the triangle is acute.
* If then the triangle is obtuse.
Edsger W. Dijkstra has stated this proposition about acute, right, and obtuse triangles in this language:
:
where ''α'' is the angle opposite to side ''a'', ''β'' is the angle opposite to side ''b'', ''γ'' is the angle opposite to side ''c'', and sgn is the sign function
In mathematics, the sign function or signum function (from '' signum'', Latin for "sign") is an odd mathematical function that extracts the sign of a real number. In mathematical expressions the sign function is often represented as . To avoi ...
.
Consequences and uses of the theorem
Pythagorean triples
A Pythagorean triple has three positive integers ''a'', ''b'', and ''c'', such that In other words, a Pythagorean triple represents the lengths of the sides of a right triangle where all three sides have integer lengths.[ Such a triple is commonly written Some well-known examples are and
A primitive Pythagorean triple is one in which ''a'', ''b'' and ''c'' are ]coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
(the greatest common divisor
In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...
of ''a'', ''b'' and ''c'' is 1).
The following is a list of primitive Pythagorean triples with values less than 100:
:(3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61), (12, 35, 37), (13, 84, 85), (16, 63, 65), (20, 21, 29), (28, 45, 53), (33, 56, 65), (36, 77, 85), (39, 80, 89), (48, 55, 73), (65, 72, 97)
Inverse Pythagorean theorem
Given a right triangle
A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right an ...
with sides and altitude
Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
(a line from the right angle and perpendicular to the hypotenuse
In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equa ...
). The Pythagorean theorem has,
:
while the inverse Pythagorean theorem
In geometry, the inverse Pythagorean theorem (also known as the reciprocal Pythagorean theorem or the upside down Pythagorean theorem) is as follows:
:Let ''A'', ''B'' be the endpoints of the hypotenuse of a right triangle ''ABC''. Let ''D'' be t ...
relates the two legs
A leg is a weight-bearing and locomotive anatomical structure, usually having a columnar shape. During locomotion, legs function as "extensible struts". The combination of movements at all joints can be modeled as a single, linear element ...
to the altitude ,
:
The equation can be transformed to,
:
where for any non-zero real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
. If the are to be integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s, the smallest solution is then
:
using the smallest Pythagorean triple . The reciprocal Pythagorean theorem is a special case of the optic equation
In number theory, the optic equation is an equation that requires the sum of the reciprocals of two positive integers ''a'' and ''b'' to equal the reciprocal of a third positive integer ''c'':Dickson, L. E., ''History of the Theory of Numbers, V ...
:
where the denominators are squares and also for a heptagonal triangle
A heptagonal triangle is an obtuse scalene triangle whose vertices coincide with the first, second, and fourth vertices of a regular heptagon (from an arbitrary starting vertex). Thus its sides coincide with one side and the adjacent shorter an ...
whose sides are square numbers.
Incommensurable lengths
One of the consequences of the Pythagorean theorem is that line segments whose lengths are incommensurable (so the ratio of which is not a rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
) can be constructed using a straightedge and compass. Pythagoras' theorem enables construction of incommensurable lengths because the hypotenuse of a triangle is related to the sides by the square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
E ...
operation.
The figure on the right shows how to construct line segments whose lengths are in the ratio of the square root of any positive integer. Each triangle has a side (labeled "1") that is the chosen unit for measurement. In each right triangle, Pythagoras' theorem establishes the length of the hypotenuse in terms of this unit. If a hypotenuse is related to the unit by the square root of a positive integer that is not a perfect square, it is a realization of a length incommensurable with the unit, such as , , . For more detail, see Quadratic irrational In mathematics, a quadratic irrational number (also known as a quadratic irrational, a quadratic irrationality or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducibl ...
.
Incommensurable lengths conflicted with the Pythagorean school's concept of numbers as only whole numbers. The Pythagorean school dealt with proportions by comparison of integer multiples of a common subunit.[
] According to one legend, Hippasus of Metapontum
Hippasus of Metapontum (; grc-gre, Ἵππασος ὁ Μεταποντῖνος, ''Híppasos''; c. 530 – c. 450 BC) was a Greek philosopher and early follower of Pythagoras. Little is known about his life or his beliefs, but he is sometimes c ...
(''ca.'' 470 B.C.) was drowned at sea for making known the existence of the irrational or incommensurable.[
; Hippasus was on a voyage at the time, and his fellows cast him overboard. See
][
A careful discussion of Hippasus's contributions is found in
]
Complex numbers
For any complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
:
the absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
or modulus is given by
:
So the three quantities, ''r'', ''x'' and ''y'' are related by the Pythagorean equation,
:
Note that ''r'' is defined to be a positive number or zero but ''x'' and ''y'' can be negative as well as positive. Geometrically ''r'' is the distance of the ''z'' from zero or the origin ''O'' in the complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
.
This can be generalised to find the distance between two points, ''z''1 and ''z''2 say. The required distance is given by
:
so again they are related by a version of the Pythagorean equation,
:
Euclidean distance
The distance formula in Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
is derived from the Pythagorean theorem. If and are points in the plane, then the distance between them, also called the Euclidean distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points.
It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefor ...
, is given by
:
More generally, in Euclidean ''n''-space, the Euclidean distance between two points, and , is defined, by generalization of the Pythagorean theorem, as:
:
If instead of Euclidean distance, the square of this value (the squared Euclidean distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two Point (geometry), points.
It can be calculated from the Cartesian coordinates of the points using the Pythagorean theo ...
, or SED) is used, the resulting equation avoids square roots and is simply a sum of the SED of the coordinates:
:
The squared form is a smooth, convex function
In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of a function, graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigra ...
of both points, and is widely used in optimization theory
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
and statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, forming the basis of least squares
The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the res ...
.
Euclidean distance in other coordinate systems
If Cartesian coordinates are not used, for example, if polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
are used in two dimensions or, in more general terms, if curvilinear coordinates
In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally inve ...
are used, the formulas expressing the Euclidean distance are more complicated than the Pythagorean theorem, but can be derived from it. A typical example where the straight-line distance between two points is converted to curvilinear coordinates can be found in the applications of Legendre polynomials in physics. The formulas can be discovered by using Pythagoras' theorem with the equations relating the curvilinear coordinates to Cartesian coordinates. For example, the polar coordinates can be introduced as:
:
Then two points with locations and are separated by a distance ''s'':
:
Performing the squares and combining terms, the Pythagorean formula for distance in Cartesian coordinates produces the separation in polar coordinates as:
:
using the trigonometric product-to-sum formulas. This formula is the law of cosines
In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states ...
, sometimes called the generalized Pythagorean theorem. From this result, for the case where the radii to the two locations are at right angles, the enclosed angle and the form corresponding to Pythagoras' theorem is regained: The Pythagorean theorem, valid for right triangles, therefore is a special case of the more general law of cosines, valid for arbitrary triangles.
Pythagorean trigonometric identity
In a right triangle with sides ''a'', ''b'' and hypotenuse ''c'', trigonometry
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. T ...
determines the sine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...
and cosine of the angle ''θ'' between side ''a'' and the hypotenuse as:
:
From that it follows:
:
where the last step applies Pythagoras' theorem. This relation between sine and cosine is sometimes called the fundamental Pythagorean trigonometric identity.[
] In similar triangles, the ratios of the sides are the same regardless of the size of the triangles, and depend upon the angles. Consequently, in the figure, the triangle with hypotenuse of unit size has opposite side of size sin ''θ'' and adjacent side of size cos ''θ'' in units of the hypotenuse.
Relation to the cross product
The Pythagorean theorem relates the cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
and dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
in a similar way:[
]
:
This can be seen from the definitions of the cross product and dot product, as
:
with n a unit vector normal to both a and b. The relationship follows from these definitions and the Pythagorean trigonometric identity.
This can also be used to define the cross product. By rearranging the following equation is obtained
:
This can be considered as a condition on the cross product and so part of its definition, for example in seven dimensions.[
][
]
Generalizations
Similar figures on the three sides
The Pythagorean theorem generalizes beyond the areas of squares on the three sides to any similar figures
In Euclidean geometry, two objects are similar if they have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly wi ...
. This was known by Hippocrates of Chios
Hippocrates of Chios ( grc-gre, Ἱπποκράτης ὁ Χῖος; c. 470 – c. 410 BC) was an ancient Greek mathematician, geometer, and astronomer.
He was born on the isle of Chios, where he was originally a merchant. After some misadve ...
in the 5th century BC, and was included by Euclid
Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
in his '' Elements'':[Euclid's ''Elements'': Book VI, Proposition VI 31: "In right-angled triangles the figure on the side subtending the right angle is equal to the similar and similarly described figures on the sides containing the right angle."]
If one erects similar figures (see Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
) with corresponding sides on the sides of a right triangle, then the sum of the areas of the ones on the two smaller sides equals the area of the one on the larger side.
This extension assumes that the sides of the original triangle are the corresponding sides of the three congruent figures (so the common ratios of sides between the similar figures are ''a:b:c'').[Putz, John F. and Sipka, Timothy A. "On generalizing the Pythagorean theorem", ''The College Mathematics Journal'' 34 (4), September 2003, pp. 291–295.] While Euclid's proof only applied to convex polygons, the theorem also applies to concave polygons and even to similar figures that have curved boundaries (but still with part of a figure's boundary being the side of the original triangle).[
The basic idea behind this generalization is that the area of a plane figure is proportional to the square of any linear dimension, and in particular is proportional to the square of the length of any side. Thus, if similar figures with areas ''A'', ''B'' and ''C'' are erected on sides with corresponding lengths ''a'', ''b'' and ''c'' then:
:
:
But, by the Pythagorean theorem, ''a''2 + ''b''2 = ''c''2, so ''A'' + ''B'' = ''C''.
Conversely, if we can prove that ''A'' + ''B'' = ''C'' for three similar figures without using the Pythagorean theorem, then we can work backwards to construct a proof of the theorem. For example, the starting center triangle can be replicated and used as a triangle ''C'' on its hypotenuse, and two similar right triangles (''A'' and ''B'' ) constructed on the other two sides, formed by dividing the central triangle by its ]altitude
Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
. The sum of the areas of the two smaller triangles therefore is that of the third, thus ''A'' + ''B'' = ''C'' and reversing the above logic leads to the Pythagorean theorem ''a''2 + ''b''2 = ''c''2. (''See also Einstein's proof by dissection without rearrangement'')
Law of cosines
The Pythagorean theorem is a special case of the more general theorem relating the lengths of sides in any triangle, the law of cosines:[
]
::
where is the angle between sides and .
When is radians or 90°, then , and the formula reduces to the usual Pythagorean theorem.
Arbitrary triangle
At any selected angle of a general triangle of sides ''a, b, c'', inscribe an isosceles triangle such that the equal angles at its base θ are the same as the selected angle. Suppose the selected angle θ is opposite the side labeled ''c''. Inscribing the isosceles triangle forms triangle ''CAD'' with angle θ opposite side ''b'' and with side ''r'' along ''c''. A second triangle is formed with angle θ opposite side ''a'' and a side with length ''s'' along ''c'', as shown in the figure. Thābit ibn Qurra
Thābit ibn Qurra (full name: , ar, أبو الحسن ثابت بن قرة بن زهرون الحراني الصابئ, la, Thebit/Thebith/Tebit); 826 or 836 – February 19, 901, was a mathematician, physician, astronomer, and translator who ...
stated that the sides of the three triangles were related as:[
][
]
:
As the angle θ approaches /2, the base of the isosceles triangle narrows, and lengths ''r'' and ''s'' overlap less and less. When θ = /2, ''ADB'' becomes a right triangle, ''r'' + ''s'' = ''c'', and the original Pythagorean theorem is regained.
One proof observes that triangle ''ABC'' has the same angles as triangle ''CAD'', but in opposite order. (The two triangles share the angle at vertex A, both contain the angle θ, and so also have the same third angle by the triangle postulate
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non-collinear ...
.) Consequently, ''ABC'' is similar to the reflection of ''CAD'', the triangle ''DAC'' in the lower panel. Taking the ratio of sides opposite and adjacent to θ,
:
Likewise, for the reflection of the other triangle,
:
Clearing fractions In mathematics, the method of clearing denominators, also called clearing fractions, is a technique for simplifying an equation equating two expressions that each are a sum of rational expressions – which includes simple fractions.
Example
Co ...
and adding these two relations:
:
the required result.
The theorem remains valid if the angle is obtuse so the lengths ''r'' and ''s'' are non-overlapping.
General triangles using parallelograms
Pappus's area theorem
Pappus's area theorem describes the relationship between the areas of three parallelograms attached to three sides of an arbitrary triangle. The theorem, which can also be thought of as a generalization of the Pythagorean theorem, is named after t ...
is a further generalization, that applies to triangles that are not right triangles, using parallelograms on the three sides in place of squares (squares are a special case, of course). The upper figure shows that for a scalene triangle, the area of the parallelogram on the longest side is the sum of the areas of the parallelograms on the other two sides, provided the parallelogram on the long side is constructed as indicated (the dimensions labeled with arrows are the same, and determine the sides of the bottom parallelogram). This replacement of squares with parallelograms bears a clear resemblance to the original Pythagoras' theorem, and was considered a generalization by Pappus of Alexandria
Pappus of Alexandria (; grc-gre, Πάππος ὁ Ἀλεξανδρεύς; AD) was one of the last great Greek mathematicians of antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem i ...
in 4 AD[For the details of such a construction, see ]
The lower figure shows the elements of the proof. Focus on the left side of the figure. The left green parallelogram has the same area as the left, blue portion of the bottom parallelogram because both have the same base ''b'' and height ''h''. However, the left green parallelogram also has the same area as the left green parallelogram of the upper figure, because they have the same base (the upper left side of the triangle) and the same height normal to that side of the triangle. Repeating the argument for the right side of the figure, the bottom parallelogram has the same area as the sum of the two green parallelograms.
Solid geometry
In terms of solid geometry
In mathematics, solid geometry or stereometry is the traditional name for the geometry of Three-dimensional space, three-dimensional, Euclidean spaces (i.e., 3D geometry).
Stereometry deals with the measurements of volumes of various solid fig ...
, Pythagoras' theorem can be applied to three dimensions as follows. Consider a rectangular solid as shown in the figure. The length of diagonal ''BD'' is found from Pythagoras' theorem as:
:
where these three sides form a right triangle. Using horizontal diagonal ''BD'' and the vertical edge ''AB'', the length of diagonal ''AD'' then is found by a second application of Pythagoras' theorem as:
:
or, doing it all in one step:
:
This result is the three-dimensional expression for the magnitude of a vector v (the diagonal AD) in terms of its orthogonal components (the three mutually perpendicular sides):
:
This one-step formulation may be viewed as a generalization of Pythagoras' theorem to higher dimensions. However, this result is really just the repeated application of the original Pythagoras' theorem to a succession of right triangles in a sequence of orthogonal planes.
A substantial generalization of the Pythagorean theorem to three dimensions is de Gua's theorem, named for Jean Paul de Gua de Malves
Jean Paul de Gua de Malves (1713, Malves-en-Minervois (Aude) – June 2, 1785, Paris) was a French mathematician who published in 1740 a work on analytical geometry in which he applied it, without the aid of differential calculus, to find the tange ...
: If a tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
has a right angle corner (like a corner of a cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only r ...
), then the square of the area of the face opposite the right angle corner is the sum of the squares of the areas of the other three faces. This result can be generalized as in the "''n''-dimensional Pythagorean theorem":[
]
This statement is illustrated in three dimensions by the tetrahedron in the figure. The "hypotenuse" is the base of the tetrahedron at the back of the figure, and the "legs" are the three sides emanating from the vertex in the foreground. As the depth of the base from the vertex increases, the area of the "legs" increases, while that of the base is fixed. The theorem suggests that when this depth is at the value creating a right vertex, the generalization of Pythagoras' theorem applies. In a different wording:[
For an extended discussion of this generalization, see, for example]
Willie W. Wong
2002, ''A generalized n-dimensional Pythagorean theorem''.
Inner product spaces
The Pythagorean theorem can be generalized to inner product space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often den ...
s,[
] which are generalizations of the familiar 2-dimensional and 3-dimensional Euclidean spaces
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
. For example, a function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
may be considered as a vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
with infinitely many components in an inner product space, as in functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
.[
]
In an inner product space, the concept of perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
ity is replaced by the concept of orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
ity: two vectors v and w are orthogonal if their inner product is zero. The inner product
In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
is a generalization of the dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
of vectors. The dot product is called the ''standard'' inner product or the ''Euclidean'' inner product. However, other inner products are possible.[
]
The concept of length is replaced by the concept of the norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
, , v, , of a vector v, defined as:[
]
:
In an inner-product space, the Pythagorean theorem states that for any two orthogonal vectors v and w we have
:
Here the vectors v and w are akin to the sides of a right triangle with hypotenuse given by the vector sum
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors a ...
v + w. This form of the Pythagorean theorem is a consequence of the properties of the inner product:
:
where because of orthogonality.
A further generalization of the Pythagorean theorem in an inner product space to non-orthogonal vectors is the ''parallelogram law
In mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the s ...
'' :[
:
which says that twice the sum of the squares of the lengths of the sides of a parallelogram is the sum of the squares of the lengths of the diagonals. Any norm that satisfies this equality is '']ipso facto
is a Latin phrase, directly translated as "by the fact itself", which means that a specific phenomenon is a ''direct'' consequence, a resultant ''effect'', of the action in question, instead of being brought about by a previous action. It is a ...
'' a norm corresponding to an inner product.[
The Pythagorean identity can be extended to sums of more than two orthogonal vectors. If v''1'', v''2'', ..., v''n'' are pairwise-orthogonal vectors in an inner-product space, then application of the Pythagorean theorem to successive pairs of these vectors (as described for 3-dimensions in the section on ]solid geometry
In mathematics, solid geometry or stereometry is the traditional name for the geometry of Three-dimensional space, three-dimensional, Euclidean spaces (i.e., 3D geometry).
Stereometry deals with the measurements of volumes of various solid fig ...
) results in the equation
:
Sets of ''m''-dimensional objects in ''n''-dimensional space
Another generalization of the Pythagorean theorem applies to Lebesgue-measurable
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
sets of objects in any number of dimensions. Specifically, the square of the measure of an ''m''-dimensional set of objects in one or more parallel ''m''-dimensional flats
Flat or flats may refer to:
Architecture
* Flat (housing), an apartment in the United Kingdom, Ireland, Australia and other Commonwealth countries
Arts and entertainment
* Flat (music), a symbol () which denotes a lower pitch
* Flat (soldier), ...
in ''n''-dimensional Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
is equal to the sum of the squares of the measures of the orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
projections of the object(s) onto all ''m''-dimensional coordinate subspaces.[
]
In mathematical terms:
:
where:
* is a measure in ''m''-dimensions (a length in one dimension, an area in two dimensions, a volume in three dimensions, etc.).
* is a set of one or more non-overlapping ''m''-dimensional objects in one or more parallel ''m''-dimensional flats in ''n''-dimensional Euclidean space.
* is the total measure (sum) of the set of ''m''-dimensional objects.
* represents an ''m''-dimensional projection of the original set onto an orthogonal coordinate subspace.
* is the measure of the ''m''-dimensional set projection onto ''m''-dimensional coordinate subspace . Because object projections can overlap on a coordinate subspace, the measure of each object projection in the set must be calculated individually, then measures of all projections added together to provide the total measure for the set of projections on the given coordinate subspace.
* is the number of orthogonal, ''m''-dimensional coordinate subspaces in ''n''-dimensional space () onto which the ''m''-dimensional objects are projected (''m'' ≤ ''n''):
Non-Euclidean geometry
The Pythagorean theorem is derived from the axioms
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
of Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
, and in fact, were the Pythagorean theorem to fail for some right triangle, then the plane in which this triangle is contained cannot be Euclidean. More precisely, the Pythagorean theorem implies, and is implied by, Euclid's Parallel (Fifth) Postulate.[
][
] Thus, right triangles in a non-Euclidean geometry
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...
[
]
do not satisfy the Pythagorean theorem. For example, in spherical geometry
300px, A sphere with a spherical triangle on it.
Spherical geometry is the geometry of the two-dimensional surface of a sphere. In this context the word "sphere" refers only to the 2-dimensional surface and other terms like "ball" or "solid sp ...
, all three sides of the right triangle (say ''a'', ''b'', and ''c'') bounding an octant of the unit sphere have length equal to /2, and all its angles are right angles, which violates the Pythagorean theorem because
.
Here two cases of non-Euclidean geometry are considered—spherical geometry
300px, A sphere with a spherical triangle on it.
Spherical geometry is the geometry of the two-dimensional surface of a sphere. In this context the word "sphere" refers only to the 2-dimensional surface and other terms like "ball" or "solid sp ...
and hyperbolic plane geometry; in each case, as in the Euclidean case for non-right triangles, the result replacing the Pythagorean theorem follows from the appropriate law of cosines.
However, the Pythagorean theorem remains true in hyperbolic geometry and elliptic geometry if the condition that the triangle be right is replaced with the condition that two of the angles sum to the third, say ''A''+''B'' = ''C''. The sides are then related as follows: the sum of the areas of the circles with diameters ''a'' and ''b'' equals the area of the circle with diameter ''c''.
Spherical geometry
For any right triangle on a sphere of radius ''R'' (for example, if γ in the figure is a right angle), with sides ''a'', ''b'', ''c'', the relation between the sides takes the form:[
]
:
This equation can be derived as a special case of the spherical law of cosines In spherical trigonometry, the law of cosines (also called the cosine rule for sides) is a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry.
Given a unit sphere, a "sphe ...
that applies to all spherical triangles:
:
By expressing the Maclaurin series
Maclaurin or MacLaurin is a surname. Notable people with the surname include:
* Colin Maclaurin (1698–1746), Scottish mathematician
* Normand MacLaurin (1835–1914), Australian politician and university administrator
* Henry Normand MacLaurin ( ...
for the cosine function as an asymptotic expansion In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to ...
with the remainder term in big O notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Lan ...
,
:
it can be shown that as the radius ''R'' approaches infinity and the arguments ''a/R'', ''b/R'', and ''c/R'' tend to zero, the spherical relation between the sides of a right triangle approaches the Euclidean form of the Pythagorean theorem. Substituting the asymptotic expansion for each of the cosines into the spherical relation for a right triangle yields
:
The constants ''a''4, ''b''4, and ''c''4 have been absorbed into the big ''O'' remainder terms since they are independent of the radius ''R''. This asymptotic relationship can be further simplified by multiplying out the bracketed quantities, cancelling the ones, multiplying through by −2, and collecting all the error terms together:
:
After multiplying through by ''R''2, the Euclidean Pythagorean relationship ''c''2 = ''a''2 + ''b''2 is recovered in the limit as the radius ''R'' approaches infinity (since the remainder term tends to zero):
:
For small right triangles (''a'', ''b'' << ''R''), the cosines can be eliminated to avoid loss of significance
In numerical analysis, catastrophic cancellation is the phenomenon that subtracting good approximations to two nearby numbers may yield a very bad approximation to the difference of the original numbers.
For example, if there are two studs, one L_ ...
, giving
:
Hyperbolic geometry
In a hyperbolic
Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry.
The following phenomena are described as ''hyperbolic'' because they ...
space with uniform Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point:
K = \kappa_1 \kappa_2.
The Gaussian radius of curvature is the reciprocal of .
F ...
−1/''R''2, for a right triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, an ...
with legs ''a'', ''b'', and hypotenuse ''c'', the relation between the sides takes the form:[
]
:
where cosh is the hyperbolic cosine
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the u ...
. This formula is a special form of the hyperbolic law of cosines In hyperbolic geometry, the "law of cosines" is a pair of theorems relating the sides and angles of triangles on a hyperbolic plane, analogous to the planar law of cosines from plane trigonometry, or the spherical law of cosines in spherical trigono ...
that applies to all hyperbolic triangles:[
]
:
with γ the angle at the vertex opposite the side ''c''.
By using the Maclaurin series
Maclaurin or MacLaurin is a surname. Notable people with the surname include:
* Colin Maclaurin (1698–1746), Scottish mathematician
* Normand MacLaurin (1835–1914), Australian politician and university administrator
* Henry Normand MacLaurin ( ...
for the hyperbolic cosine, , it can be shown that as a hyperbolic triangle becomes very small (that is, as ''a'', ''b'', and ''c'' all approach zero), the hyperbolic relation for a right triangle approaches the form of Pythagoras' theorem.
For small right triangles (''a'', ''b'' << ''R''), the hyperbolic cosines can be eliminated to avoid loss of significance
In numerical analysis, catastrophic cancellation is the phenomenon that subtracting good approximations to two nearby numbers may yield a very bad approximation to the difference of the original numbers.
For example, if there are two studs, one L_ ...
, giving
:
Very small triangles
For any uniform curvature ''K'' (positive, zero, or negative), in very small right triangles (, ''K'', ''a''2, , ''K'', ''b''2 << 1) with hypotenuse ''c'', it can be shown that
:
Differential geometry
The Pythagorean theorem applies to infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
triangles seen in differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
. In three dimensional space, the distance between two infinitesimally separated points satisfies
:
with ''ds'' the element of distance and (''dx'', ''dy'', ''dz'') the components of the vector separating the two points. Such a space is called a Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
. However, in Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to poin ...
, a generalization of this expression useful for general coordinates (not just Cartesian) and general spaces (not just Euclidean) takes the form:[
]
:
which is called the metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
. (Sometimes, by abuse of language, the same term is applied to the set of coefficients .) It may be a function of position, and often describes curved space
Curved space often refers to a spatial geometry which is not "flat", where a flat space is described by Euclidean geometry. Curved spaces can generally be described by Riemannian geometry though some simple cases can be described in other ways. Cu ...
. A simple example is Euclidean (flat) space expressed in curvilinear coordinates
In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally inve ...
. For example, in polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
:
:
History
There is debate whether the Pythagorean theorem was discovered once, or many times in many places, and the date of first discovery is uncertain, as is the date of the first proof. Historians of Mesopotamian
Mesopotamia ''Mesopotamíā''; ar, بِلَاد ٱلرَّافِدَيْن or ; syc, ܐܪܡ ܢܗܪ̈ܝܢ, or , ) is a historical region of Western Asia situated within the Tigris–Euphrates river system, in the northern part of the F ...
mathematics have concluded that the Pythagorean rule was in widespread use during the Old Babylonian period
The Old Babylonian Empire, or First Babylonian Empire, is dated to BC – BC, and comes after the end of Sumerian power with the destruction of the Third Dynasty of Ur, and the subsequent Isin-Larsa period. The chronology of the first dynasty ...
(20th to 16th centuries BC), over a thousand years before Pythagoras was born. The history of the theorem can be divided into four parts: knowledge of Pythagorean triples
A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...
, knowledge of the relationship among the sides of a right triangle, knowledge of the relationships among adjacent angles, and proofs of the theorem within some deductive system
A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system.
A form ...
.
Written between 2000 and 1786 BC, the Egypt
Egypt ( ar, مصر , ), officially the Arab Republic of Egypt, is a transcontinental country spanning the northeast corner of Africa and southwest corner of Asia via a land bridge formed by the Sinai Peninsula. It is bordered by the Mediter ...
ian Middle Kingdom ''Berlin Papyrus 6619
The Berlin Papyrus 6619, simply called the Berlin Papyrus when the context makes it clear, is one of the primary sources of ancient Egyptian mathematics. One of the two mathematics problems on the Papyrus may suggest that the ancient Egyptians kn ...
'' includes a problem whose solution is the Pythagorean triple 6:8:10, but the problem does not mention a triangle. The Mesopotamian tablet ''Plimpton 322
Plimpton 322 is a Babylonian clay tablet, notable as containing an example of Babylonian mathematics. It has number 322 in the G.A. Plimpton Collection at Columbia University. This tablet, believed to have been written about 1800 BC, has a table ...
'', written between 1790 and 1750 BC during the reign of King Hammurabi
Hammurabi (Akkadian: ; ) was the sixth Amorite king of the Old Babylonian Empire, reigning from to BC. He was preceded by his father, Sin-Muballit, who abdicated due to failing health. During his reign, he conquered Elam and the city-states ...
the Great, contains many entries closely related to Pythagorean triples.
In India
India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on the so ...
, the ''Baudhayana
The (Sanskrit: बौधायन) are a group of Vedic Sanskrit texts which cover dharma, daily ritual, mathematics and is one of the oldest Dharma-related texts of Hinduism that have survived into the modern age from the 1st-millennium BCE. Th ...
Shulba Sutra The ''Shulva Sutras'' or ''Śulbasūtras'' (Sanskrit: शुल्बसूत्र; ': "string, cord, rope") are sutra texts belonging to the Śrauta ritual and containing geometry related to fire-altar construction.
Purpose and origins
The ...
'', the dates of which are given variously as between the 8th and 5th century BC, contains a list of Pythagorean triples and a statement of the Pythagorean theorem, both in the special case of the isosceles
In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
right triangle
A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right an ...
and in the general case, as does the ''Apastamba
''Āpastamba Dharmasūtra'' (Sanskrit: आपस्तम्ब धर्मसूत्र) is a Sanskrit text and one of the oldest Dharma-related texts of Hinduism that have survived into the modern age from the 1st-millennium BCE. It is one o ...
Shulba Sutra'' (c. 600 BC).
Byzantine
The Byzantine Empire, also referred to as the Eastern Roman Empire or Byzantium, was the continuation of the Roman Empire primarily in its eastern provinces during Late Antiquity and the Middle Ages, when its capital city was Constantinopl ...
Neoplatonic
Neoplatonism is a strand of Platonic philosophy that emerged in the 3rd century AD against the background of Hellenistic philosophy and religion. The term does not encapsulate a set of ideas as much as a chain of thinkers. But there are some ide ...
philosopher and mathematician Proclus
Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor ( grc-gre, Πρόκλος ὁ Διάδοχος, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophers ...
, writing in the fifth century AD, states two arithmetic rules, "one of them attributed to Plato
Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
, the other to Pythagoras", for generating special Pythagorean triples. The rule attributed to Pythagoras () starts from an odd number
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because
\begin
-2 \cdot 2 &= -4 \\
0 \cdot 2 &= 0 \\
41 ...
and produces a triple with leg and hypotenuse differing by one unit; the rule attributed to Plato (428/427 or 424/423 – 348/347 BC) starts from an even number and produces a triple with leg and hypotenuse differing by two units. According to Thomas L. Heath (1861–1940), no specific attribution of the theorem to Pythagoras exists in the surviving Greek literature from the five centuries after Pythagoras lived.[page 351]
/ref> However, when authors such as Plutarch
Plutarch (; grc-gre, Πλούταρχος, ''Ploútarchos''; ; – after AD 119) was a Greek Middle Platonist philosopher, historian, biographer, essayist, and priest at the Temple of Apollo in Delphi. He is known primarily for his ''P ...
and Cicero
Marcus Tullius Cicero ( ; ; 3 January 106 BC – 7 December 43 BC) was a Roman statesman, lawyer, scholar, philosopher, and academic skeptic, who tried to uphold optimate principles during the political crises that led to the estab ...
attributed the theorem to Pythagoras, they did so in a way which suggests that the attribution was widely known and undoubted.[: "Though this is the proposition universally associated by tradition with the name of Pythagoras, no really trustworthy evidence exists that it was actually discovered by him. The comparatively late writers who attribute it to him add the story that he sacrificed an ox to celebrate his discovery."][
An extensive discussion of the historical evidence is provided in]
page=351
/ref> Classicist
Classics or classical studies is the study of classical antiquity. In the Western world, classics traditionally refers to the study of Classical Greek and Roman literature and their related original languages, Ancient Greek and Latin. Classics ...
Kurt von Fritz
Karl Albert Kurt von Fritz (25 August 1900 in Metz – 16 July 1985 in Feldafing) was a German classical philologist.
Appointed to an extraordinary professorship for Greek at the University of Rostock in 1933, he was one of the two German profes ...
wrote, "Whether this formula is rightly attributed to Pythagoras personally, but one can safely assume that it belongs to the very oldest period of Pythagorean mathematics."[ Around 300 BC, in Euclid's ''Elements'', the oldest extant axiomatic proof of the theorem is presented.][
]
With contents known much earlier, but in surviving texts dating from roughly the 1st century BC, the Chinese
Chinese can refer to:
* Something related to China
* Chinese people, people of Chinese nationality, citizenship, and/or ethnicity
**''Zhonghua minzu'', the supra-ethnic concept of the Chinese nation
** List of ethnic groups in China, people of ...
text ''Zhoubi Suanjing
The ''Zhoubi Suanjing'' () is one of the oldest Chinese mathematical texts. "Zhou" refers to the ancient Zhou dynasty (1046–256 BCE); "Bì" literally means "thigh", but in the book refers to the gnomon of a sundial. The book is dedicated to ...
'' (周髀算经), (''The Arithmetical Classic of the Gnomon
A gnomon (; ) is the part of a sundial that casts a shadow. The term is used for a variety of purposes in mathematics and other fields.
History
A painted stick dating from 2300 BC that was excavated at the astronomical site of Taosi is the ol ...
and the Circular Paths of Heaven'') gives a reasoning for the Pythagorean theorem for the (3, 4, 5) triangle — in China it is called the "Gougu theorem" (勾股定理).[
][
A rather extensive discussion of the origins of the various texts in the Zhou Bi is provided by
] During the Han Dynasty
The Han dynasty (, ; ) was an imperial dynasty of China (202 BC – 9 AD, 25–220 AD), established by Liu Bang (Emperor Gao) and ruled by the House of Liu. The dynasty was preceded by the short-lived Qin dynasty (221–207 BC) and a warr ...
(202 BC to 220 AD), Pythagorean triples appear in ''The Nine Chapters on the Mathematical Art
''The Nine Chapters on the Mathematical Art'' () is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 2nd century CE. This book is one of the earliest sur ...
'',[
This work is a compilation of 246 problems, some of which survived the book burning of 213 BC, and was put in final form before 100 AD. It was extensively commented upon by Liu Hui in 263 AD. See particularly §3: ''Nine chapters on the mathematical art'', pp. 71 ''ff''.
] together with a mention of right triangles.[
] Some believe the theorem arose first in China
China, officially the People's Republic of China (PRC), is a country in East Asia. It is the world's most populous country, with a population exceeding 1.4 billion, slightly ahead of India. China spans the equivalent of five time zones and ...
,[
In particular, Li Jimin; see
] where it is alternatively known as the "Shang Gao theorem" (商高定理),[
] named after the Duke of Zhou's astronomer and mathematician, whose reasoning composed most of what was in the ''Zhoubi Suanjing''.[
]
See also
*Addition in quadrature
In mathematics, Pythagorean addition is a binary operation on the real numbers that computes the length of the hypotenuse of a right triangle, given its two sides. According to the Pythagorean theorem, for a triangle with sides a and b, this lengt ...
*At Dulcarnon ''Dulcarnon'' or ''dulcarnoun'' is a term used in the Middle English poem ''Troilus and Criseyde'' by Geoffrey Chaucer, in a line given to Criseyde: "at dulcarnoun, right at my wittes ende". It became proverbial. The etymology is from an Arabic ph ...
*British flag theorem
In Euclidean geometry, the British flag theorem says that if a point ''P'' is chosen inside a rectangle ''ABCD'' then the sum of the squares of the Euclidean distances from ''P'' to two opposite corners of the rectangle equals the sum to the oth ...
*Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been k ...
*Inverse Pythagorean theorem
In geometry, the inverse Pythagorean theorem (also known as the reciprocal Pythagorean theorem or the upside down Pythagorean theorem) is as follows:
:Let ''A'', ''B'' be the endpoints of the hypotenuse of a right triangle ''ABC''. Let ''D'' be t ...
*Kepler triangle
A Kepler triangle is a special right triangle with edge lengths in geometric progression. The ratio of the progression is \sqrt\varphi where \varphi=(1+\sqrt)/2 is the golden ratio, and the progression can be written: or approximately . Squares ...
*Linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
*List of triangle topics
This list of triangle topics includes things related to the geometric shape, either abstractly, as in idealizations studied by geometers, or in triangular arrays such as Pascal's triangle or triangular matrix, triangular matrices, or concretely in ...
* L''p'' space
*Nonhypotenuse number
In mathematics, a nonhypotenuse number is a natural number whose square ''cannot'' be written as the sum of two nonzero squares. The name stems from the fact that an edge of length equal to a nonhypotenuse number ''cannot'' form the hypotenuse of ...
*Parallelogram law
In mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the s ...
*Parseval's identity In mathematical analysis, Parseval's identity, named after Marc-Antoine Parseval, is a fundamental result on the summability of the Fourier series of a function. Geometrically, it is a generalized Pythagorean theorem for inner-product spaces (which ...
*Ptolemy's theorem
In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle). The theorem is named after the Greek astronomer and mathematician ...
*Pythagorean expectation
Pythagorean expectation is a sports analytics formula devised by Bill James to estimate the percentage of games a baseball team "should" have won based on the number of runs they scored and allowed. Comparing a team's actual and Pythagorean win ...
*Pythagorean tiling
A Pythagorean tiling or two squares tessellation is a tiling of a Euclidean plane by squares of two different sizes, in which each square touches four squares of the other size on its four sides. Many proofs of the Pythagorean theorem are ...
* Rational trigonometry in Pythagoras' theorem
* Thales theorem
Notes and references
Notes
References
Works cited
*
* On-line text a
archive.org
*
* This high-school geometry text covers many of the topics in this WP article.
* For full text of 2nd edition of 1940, see Originally published in 1940 and reprinted in 1968 by National Council of Teachers of Mathematics, .
*
*
* Robson, Eleanor and Jacqueline Stedall, eds., The Oxford Handbook of the History of Mathematics, Oxford: Oxford University Press, 2009. pp. vii + 918. .
* Also .
*
*
External links
* In HTML with Java-based interactive figures.
*
* Interactive links:
*
in Java
Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's List ...
of the Pythagorean theorem
*
Another interactive proof
in Java
Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's List ...
of the Pythagorean theorem
*
Pythagorean theorem
with interactive animation
*
Pythagorean theorem
Pythagorean theorem water demo
on YouTube
Pythagorean theorem
(more than 70 proofs from cut-the-knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...
)
*
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In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...
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Theorems_in_plane_geometry