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geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, Thales's theorem states that if , , and are distinct points on a
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
where the line is a
diameter In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest Chord (geometry), chord of the circle. Both definitions a ...
, the
angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
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 ...
. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as part of the 31st proposition in the third book of
Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...
's '' Elements''. It is generally attributed to
Thales of Miletus Thales of Miletus ( ; ; ) was an Ancient Greek pre-Socratic philosopher from Miletus in Ionia, Asia Minor. Thales was one of the Seven Sages, founding figures of Ancient Greece. Beginning in eighteenth-century historiography, many came to ...
, but it is sometimes attributed to
Pythagoras Pythagoras of Samos (;  BC) was an ancient Ionian Greek philosopher, polymath, and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graecia and influenced the philosophies of P ...
.


History

Babylonian mathematicians knew this for special cases before Greek mathematicians proved it.
Thales of Miletus Thales of Miletus ( ; ; ) was an Ancient Greek pre-Socratic philosopher from Miletus in Ionia, Asia Minor. Thales was one of the Seven Sages, founding figures of Ancient Greece. Beginning in eighteenth-century historiography, many came to ...
(early 6th century BC) is traditionally credited with proving the theorem; however, even by the 5th century BC there was nothing extant of Thales' writing, and inventions and ideas were attributed to men of wisdom such as Thales and Pythagoras by later doxographers based on hearsay and speculation. Reference to Thales was made by
Proclus Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor (, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophers of late antiquity. He set forth one of th ...
(5th century AD), and by
Diogenes Laërtius Diogenes Laërtius ( ; , ; ) was a biographer of the Greek philosophers. Little is definitively known about his life, but his surviving book ''Lives and Opinions of Eminent Philosophers'' is a principal source for the history of ancient Greek ph ...
(3rd century AD) documenting Pamphila's (1st century AD) statement that Thales "was the first to inscribe in a circle a right-angle triangle". Thales was claimed to have traveled to
Egypt Egypt ( , ), officially the Arab Republic of Egypt, is a country spanning the Northeast Africa, northeast corner of Africa and Western Asia, southwest corner of Asia via the Sinai Peninsula. It is bordered by the Mediterranean Sea to northe ...
and
Babylonia Babylonia (; , ) was an Ancient history, ancient Akkadian language, Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Kuwait, Syria and Iran). It emerged as a ...
, where he is supposed to have learned about geometry and astronomy and thence brought their knowledge to the Greeks, along the way inventing the concept of geometric proof and proving various geometric theorems. However, there is no direct evidence for any of these claims, and they were most likely invented speculative rationalizations. Modern scholars believe that Greek deductive geometry as found in Euclid's ''Elements'' was not developed until the 4th century BC, and any geometric knowledge Thales may have had would have been observational. The theorem appears in Book III of Euclid's ''Elements'' () as proposition 31: "In a circle the angle in the semicircle is right, that in a greater segment less than a right angle, and that in a less segment greater than a right angle; further the angle of the greater segment is greater than a right angle, and the angle of the less segment is less than a right angle."
Dante Alighieri Dante Alighieri (; most likely baptized Durante di Alighiero degli Alighieri; – September 14, 1321), widely known mononymously as Dante, was an Italian Italian poetry, poet, writer, and philosopher. His ''Divine Comedy'', originally called ...
's '' Paradiso'' (canto 13, lines 101–102) refers to Thales's theorem in the course of a speech.


Proof


First proof

The following facts are used: the sum of the angles in a
triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
is equal to 180° and the base angles of an
isosceles triangle In geometry, an isosceles triangle () is a triangle that has two Edge (geometry), sides of equal length and two angles of equal measure. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at le ...
are equal. Since , and are isosceles triangles, and by the equality of the base angles of an isosceles triangle, and . Let and . The three internal angles of the triangle are , , and . Since the sum of the angles of a triangle is equal to 180°, we have :\begin \alpha+ (\alpha + \beta) + \beta &= 180^\circ \\ 2\alpha + 2\beta &= 180^\circ \\ 2( \alpha + \beta ) &= 180^\circ \\ \therefore \alpha + \beta &= 90^\circ. \end
Q.E.D. Q.E.D. or QED is an initialism of the List of Latin phrases (full), Latin phrase , meaning "that which was to be demonstrated". Literally, it states "what was to be shown". Traditionally, the abbreviation is placed at the end of Mathematical proof ...


Second proof

The theorem may also be proven using
trigonometry Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The fiel ...
: Let , , and . Then is a point on the unit circle . We will show that forms a right angle by proving that and are
perpendicular In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', � ...
— that is, the product of their
slope In mathematics, the slope or gradient of a Line (mathematics), line is a number that describes the direction (geometry), direction of the line on a plane (geometry), plane. Often denoted by the letter ''m'', slope is calculated as the ratio of t ...
s is equal to −1. We calculate the slopes for and : :\begin m_ &= \frac = \frac \\ ptm_ &= \frac = \frac \end Then we show that their product equals −1: : \begin &m_ \cdot m_\\ pt= & \frac \cdot \frac\\ pt= & \frac\\ pt= & \frac\\ pt= & \end Note the use of the
Pythagorean trigonometric identity The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations ...
\sin^2 \theta + \cos^2 \theta = 1.


Third proof

Let be a triangle in a circle where is a diameter in that circle. Then construct a new triangle by mirroring over the line and then mirroring it again over the line perpendicular to which goes through the center of the circle. Since lines and are parallel, likewise for and , the
quadrilateral In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...
is a
parallelogram In Euclidean geometry, a parallelogram is a simple polygon, simple (non-list of self-intersecting polygons, self-intersecting) quadrilateral with two pairs of Parallel (geometry), parallel sides. The opposite or facing sides of a parallelogram a ...
. Since lines and , the diagonals of the parallelogram, are both diameters of the circle and therefore have equal length, the parallelogram must be a rectangle. All angles in a rectangle are right angles.


Fourth proof

The theorem can be proved using vector algebra. Let's take the vectors \overrightarrow and \overrightarrow. These vectors satisfy :\overrightarrow = \overrightarrow + \overrightarrow\qquad \qquad \overrightarrow = \overrightarrow + \overrightarrow and their dot product can be expanded as :\overrightarrow\cdot\overrightarrow = \left(\overrightarrow + \overrightarrow\right)\cdot\left( \overrightarrow + \overrightarrow\right) =\overrightarrow\cdot\overrightarrow+(\overrightarrow+\overrightarrow)\cdot \overrightarrow+\overrightarrow\cdot\overrightarrow but :\overrightarrow=-\overrightarrow\qquad\qquad \overrightarrow\cdot\overrightarrow=-R^2\qquad\qquad \overrightarrow\cdot\overrightarrow = R^2 and the dot product vanishes :\overrightarrow\cdot\overrightarrow = -R^2+\overrightarrow\cdot \overrightarrow+R^2=0 and then the vectors \overrightarrow and \overrightarrow are orthogonal and the angle ABC is a right angle.


Converse

For any triangle, and, in particular, any right triangle, there is exactly one circle containing all three vertices of the triangle. This circle is called the
circumcircle In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertex (geometry), vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumrad ...
of the triangle. The locus of points equidistant from two given points is a straight line that is called the perpendicular bisector of the line segment connecting the points. The perpendicular bisectors of any two sides of a triangle intersect in exactly one point. This point must be equidistant from the vertices of the triangle. One way of formulating Thales's theorem is: if the center of a triangle's circumcircle lies on the triangle then the triangle is right, and the center of its circumcircle lies on its hypotenuse. The converse of Thales's theorem is then: the center of the circumcircle of a right triangle lies on its hypotenuse. (Equivalently, a right triangle's hypotenuse is a diameter of its circumcircle.)


Proof of the converse using geometry

This proof consists of 'completing' the right triangle to form a
rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that a ...
and noticing that the center of that rectangle is equidistant from the vertices and so is the center of the circumscribing circle of the original triangle, it utilizes two facts: *adjacent angles in a
parallelogram In Euclidean geometry, a parallelogram is a simple polygon, simple (non-list of self-intersecting polygons, self-intersecting) quadrilateral with two pairs of Parallel (geometry), parallel sides. The opposite or facing sides of a parallelogram a ...
are supplementary (add to 180°) and, *the diagonals of a rectangle are equal and cross each other in their median point. Let there be a right angle , a line parallel to passing by , and a line parallel to passing by . Let be the point of intersection of lines and . (It has not been proven that lies on the circle.) The quadrilateral forms a parallelogram by construction (as opposite sides are parallel). Since in a parallelogram adjacent angles are supplementary (add to 180°) and is a right angle (90°) then angles are also right (90°); consequently is a rectangle. Let be the point of intersection of the diagonals and . Then the point , by the second fact above, is equidistant from , , and . And so is center of the circumscribing circle, and the hypotenuse of the triangle () is a diameter of the circle.


Alternate proof of the converse using geometry

Given a right triangle with hypotenuse , construct a circle whose diameter is . Let be the center of . Let be the intersection of and the ray . By Thales's theorem, is right. But then must equal . (If lies inside , would be obtuse, and if lies outside , would be acute.)


Proof of the converse using linear algebra

This proof utilizes two facts: *two lines form a right angle if and only if the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a Scalar (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
of their directional
vector Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...
s is zero, and *the square of the length of a vector is given by the dot product of the vector with itself. Let there be a right angle and circle with as a diameter. Let M's center lie on the origin, for easier calculation. Then we know *, because the circle centered at the origin has as diameter, and *, because is a right angle. It follows :\begin 0 &= (A-B) \cdot (B-C) \\ &= (A-B) \cdot (B+A) \\ &= , A, ^2 - , B, ^2. \\ pt \therefore \ , A, &= , B, . \end This means that and are equidistant from the origin, i.e. from the center of . Since lies on , so does , and the circle is therefore the triangle's circumcircle. The above calculations in fact establish that both directions of Thales's theorem are valid in any
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, ofte ...
.


Generalizations and related results

As stated above, Thales's theorem is a special case of the inscribed angle theorem (the proof of which is quite similar to the first proof of Thales's theorem given above): :Given three points , and on a circle with center , the angle is twice as large as the angle . A related result to Thales's theorem is the following: *If is a diameter of a circle, then: :*If is inside the circle, then :*If is on the circle, then :*If is outside the circle, then .


Applications


Constructing a tangent to a circle passing through a point

Thales's theorem can be used to construct the
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
to a given circle that passes through a given point. In the figure at right, given circle with centre and the point outside , bisect at and draw the circle of radius with centre . is a diameter of this circle, so the triangles connecting OP to the points and where the circles intersect are both right triangles.


Finding the centre of a circle

Thales's theorem can also be used to find the centre of a circle using an object with a right angle, such as a
set square A set square or triangle (American English) is an object used in engineering and technical drawing, with the aim of providing a straightedge at a right angle or other particular planar angle to a baseline. Types The simplest form of set s ...
or rectangular sheet of paper larger than the circle.Resources for Teaching Mathematics: 14–16
Colin Foster The angle is placed anywhere on its circumference (figure 1). The intersections of the two sides with the circumference define a diameter (figure 2). Repeating this with a different set of intersections yields another diameter (figure 3). The centre is at the intersection of the diameters.


See also

*
Synthetic geometry Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is geometry without the use of coordinates. It relies on the axiomatic method for proving all results from a few basic properties initially called postulates ...
* Inverse Pythagorean theorem


Notes


References

* *


External links

*
Munching on Inscribed Angles


with interactive animation
Demos of Thales's theorem
by Michael Schreiber,
The Wolfram Demonstrations Project The Wolfram Demonstrations Project is an open-source collection of interactive programmes called Demonstrations. It is hosted by Wolfram Research. At its launch, it contained 1300 demonstrations but has grown to over 10,000. The site won a Pa ...
. {{Ancient Greek mathematics Euclidean plane geometry Greek mathematics Articles containing proofs Theorems about right triangles Theorems about triangles and circles es:Teorema de Tales#Segundo teorema he:משפט תאלס#המשפט השני