Thales' Theorem

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In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
, Thales's theorem states that if A, B, and C are distinct points on a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
where the line is a
diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid ...
, the
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
ABC is a
right angle In geometry and trigonometry, a right angle is an angle of exactly 90 degrees or radians corresponding to a quarter turn. If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. Th ...
. 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 (; grc-gre, Εὐκλείδης; 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 ...
's '' Elements''. It is generally attributed to
Thales of Miletus Thales of Miletus ( ; grc-gre, Θαλῆς; ) was a Greek mathematician, astronomer, statesman, and pre-Socratic philosopher from Miletus in Ionia, Asia Minor. He was one of the Seven Sages of Greece. Many, most notably Aristotle, regard ...
, but it is sometimes attributed to
Pythagoras Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samian, or simply ; in Ionian Greek; ) was an ancient Ionian Greek philosopher and the eponymous founder of Pythagoreanism. His polit ...
.

# History

There is nothing extant of the writing of
Thales Thales of Miletus ( ; grc-gre, Θαλῆς; ) was a Greek mathematician, astronomer, statesman, and pre-Socratic philosopher from Miletus in Ionia, Asia Minor. He was one of the Seven Sages of Greece. Many, most notably Aristotle, regarded hi ...
. Work done in
ancient Greece Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity ( AD 600), that comprised a loose collection of cul ...
tended to be attributed to men of wisdom without respect to all the individuals involved in any particular intellectual constructions; this is true of Pythagoras especially. Attribution did tend to occur at a later time. Reference to Thales was made by Proclus, and by
Diogenes Laërtius Diogenes Laërtius ( ; grc-gre, Διογένης Λαέρτιος, ; ) was a biographer of the Greek philosophers. Nothing is definitively known about his life, but his surviving ''Lives and Opinions of Eminent Philosophers'' is a principal sourc ...
documenting Pamphila's statement that Thales "was the first to inscribe in a circle a right-angle triangle". Babylonian mathematicians knew this for special cases before Thales proved it. It is believed that Thales learned that an angle inscribed in a
semicircle In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. The full arc of a semicircle always measures 180° (equivalently, radians, or a half-turn). It has only one line ...
is a right angle during his travels to
Babylon ''Bābili(m)'' * sux, 𒆍𒀭𒊏𒆠 * arc, 𐡁𐡁𐡋 ''Bāḇel'' * syc, ܒܒܠ ''Bāḇel'' * grc-gre, Βαβυλών ''Babylṓn'' * he, בָּבֶל ''Bāvel'' * peo, 𐎲𐎠𐎲𐎡𐎽𐎢 ''Bābiru'' * elx, 𒀸𒁀𒉿𒇷 ''Babi ...
. The theorem is named after Thales because he was said by ancient sources to have been the first to prove the theorem, using his own results that the base angles of an isosceles triangle are equal, and that the sum of angles of a triangle is equal to a straight angle (180°). Dante'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 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-colline ...
is equal to 180 ° and the base angles of an
isosceles triangle 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 ...
are equal. Since = = , ∆OBA and ∆OBC are isosceles triangles, and by the equality of the base angles of an isosceles triangle, ∠OBC = ∠OCB and ∠OBA = ∠OAB. Let ''α'' = ∠BAO and β = ∠OBC. The three internal angles of the ∆ABC triangle are ''α'', (''α'' + ''β''), and ''β''. Since the sum of the angles of a triangle is equal to 180°, we have :$\alpha+\left\left( \alpha + \beta \right\right) + \beta = 180^\circ$ :$2\alpha + 2\beta =180^\circ$ :$2\left( \alpha + \beta \right) =180^\circ$ :$\therefore \alpha + \beta =90^\circ.$
Q.E.D. Q.E.D. or QED is an initialism of the Latin phrase , meaning "which was to be demonstrated". Literally it states "what was to be shown". Traditionally, the abbreviation is placed at the end of mathematical proofs and philosophical arguments in pr ...

## Second proof

The theorem may also be proven using
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 ...
: Let $O = \left(0, 0\right)$, $A = \left(-1, 0\right)$, and $C = \left(1, 0\right)$. Then B is a point on the unit circle $\left(\cos \theta, \sin \theta\right)$. We will show that ∆ABC forms a right angle by proving that and are
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 ...
— that is, the product of their
slope In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
s is equal to −1. We calculate the slopes for and : :$m_ = \frac = \frac$ and :$m_ = \frac = \frac$ Then we show that their product equals −1: : Note the use of the Pythagorean trigonometric identity $\sin^2 \theta + \cos^2 \theta = 1$.

## Third proof

Let $ABC$ be a triangle in a circle where $AB$ is a diameter in that circle. Then construct a new triangle $ABD$ by mirroring triangle $ABC$ over the line $AB$ and then mirroring it again over the line perpendicular to $AB$ which goes through the center of the circle. Since lines $AC$ and $BD$ are parallel, likewise for $AD$ and $CB$, the
quadrilateral In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
$ACBD$ is a
parallelogram In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of eq ...
. Since lines $AB$ and $CD$, 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.

# Converse

For any triangle, and, in particular, any right triangle, there is exactly one circle containing all three vertices of the triangle. (''Sketch of proof''. 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.) This circle is called the
circumcircle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
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 plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containin ...
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 (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of eq ...
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 ∠ABC, r a line parallel to passing by A and s a line parallel to passing by C. Let D be the point of intersection of lines r and s (Note that it has not been proven that D lies on the circle) The quadrilateral ABCD forms a parallelogram by construction (as opposite sides are parallel). Since in a parallelogram adjacent angles are supplementary (add to 180°) and ∠ABC is a right angle (90°) then angles ∠BAD, ∠BCD, and ∠ADC are also right (90°); consequently ABCD is a rectangle. Let O be the point of intersection of the diagonals and . Then the point O, by the second fact above, is equidistant from A, B, and C. And so O 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 Ω whose diameter is . Let be the center of Ω. Let be the intersection of Ω 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 as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algeb ...
of their directional
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 ...
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 ∠ABC and circle M with as a diameter. Let M's center lie on the origin, for easier calculation. Then we know *A = − C, because the circle centered at the origin has as diameter, and *(A − B) · (B − C) = 0, because ∠ABC is a right angle. It follows :0 = (A − B) · (B − C) = (A − B) · (B + A) = , A, 2 − , B, 2. Hence: :, A, = , B, . This means that ''A'' and ''B'' are equidistant from the origin, i.e. from the center of ''M''. Since ''A'' lies on ''M'', so does ''B'', and the circle ''M'' 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, often ...
.

# Generalizations and related results

Thales's theorem is a special case of the following theorem: :Given three points A, B and C on a circle with center O, the angle ∠AOC is twice as large as the angle ∠ABC. See inscribed angle, the proof of this theorem is quite similar to the proof of Thales's theorem given above. A related result to Thales's theorem is the following: *If is a diameter of a circle, then: :*If B is inside the circle, then ∠ABC > 90° :*If B is on the circle, then ∠ABC = 90° :*If B is outside the circle, then ∠ABC < 90°.

# Application

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 the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
to a given circle that passes through a given point. In the figure at right, given circle ''k'' with centre O and the point P outside ''k'', bisect OP at H and draw the circle of radius OH with centre H. OP is a diameter of this circle, so the triangles connecting OP to the points T and T′ where the circles intersect are both right triangles. 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 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.

*
Synthetic geometry Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry without the use of coordinates or formulae. It relies on the axiomatic method and the tools directly related to them, that is, compas ...
* Inverse Pythagorean theorem

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