# Thales's theorem

Last updated

In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ABC is a right angle. 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's Elements . [1] It is generally attributed to Thales of Miletus, but it is sometimes attributed to Pythagoras.

## History

o se del mezzo cerchio faru si puotetriangol sì ch'un retto non avesse.

Or if in semicircle can be made
Triangle so that it have no right angle.

Dante's Paradiso, Canto 13, lines 101–102. English translation by Henry Wadsworth Longfellow.

There is nothing extant of the writing of Thales; work done in ancient Greece 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. [2] Reference to Thales was made by Proclus, and by Diogenes Laërtius documenting Pamphila's statement that Thales [3] "was the first to inscribe in a circle a right-angle triangle".

Indian and Babylonian mathematicians knew this for special cases before Thales proved it. [4] It is believed that Thales learned that an angle inscribed in a semicircle is a right angle during his travels to Babylon. [5] 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 in a triangle is equal to 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 is equal to 180° and the base angles of an isosceles triangle are equal.

Since OA = OB = OC, ∆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

${\displaystyle \alpha +\left(\alpha +\beta \right)+\beta =180^{\circ }}$
${\displaystyle 2\alpha +2\beta =180^{\circ }}$
${\displaystyle 2(\alpha +\beta )=180^{\circ }}$
${\displaystyle \therefore \alpha +\beta =90^{\circ }.}$

### Second proof

The theorem may also be proven using trigonometry: Let ${\displaystyle O=(0,0)}$, ${\displaystyle A=(-1,0)}$, and ${\displaystyle C=(1,0)}$. Then B is a point on the unit circle ${\displaystyle (\cos \theta ,\sin \theta )}$. We will show that ∆ABC forms a right angle by proving that AB and BC are perpendicular — that is, the product of their slopes is equal to −1. We calculate the slopes for AB and BC:

${\displaystyle m_{AB}={\frac {y_{B}-y_{A}}{x_{B}-x_{A}}}={\frac {\sin \theta }{\cos \theta +1}}}$

and

${\displaystyle m_{BC}={\frac {y_{B}-y_{C}}{x_{B}-x_{C}}}={\frac {\sin \theta }{\cos \theta -1}}}$

Then we show that their product equals −1:

{\displaystyle {\begin{aligned}&m_{AB}\cdot m_{BC}\\[8pt]={}&{\frac {\sin \theta }{\cos \theta +1}}\cdot {\frac {\sin \theta }{\cos \theta -1}}\\[8pt]={}&{\frac {\sin ^{2}\theta }{\cos ^{2}\theta -1}}\\[8pt]={}&{\frac {\sin ^{2}\theta }{-\sin ^{2}\theta }}\\[8pt]={}&{-1}\end{aligned}}}

Note the use of the Pythagorean trigonometric identity ${\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1}$.

### Third proof

Let ${\displaystyle ABC}$ be a triangle in a circle where ${\displaystyle AB}$ is a diameter in that circle. Then construct a new triangle ${\displaystyle ABD}$ by mirroring triangle ${\displaystyle ABC}$ over the line ${\displaystyle AB}$ and then mirroring it again over the line perpendicular to ${\displaystyle AB}$ which goes through the center of the circle. Since lines ${\displaystyle AC}$ and ${\displaystyle BD}$ are parallel, likewise for ${\displaystyle AD}$ and ${\displaystyle CB}$, the quadrilateral ${\displaystyle ACBD}$ is a parallelogram. Since lines ${\displaystyle AB}$ and ${\displaystyle CD}$, 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 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 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 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 BC passing by A and s a line parallel to AB 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 AC and BD. 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 (AC) is a diameter of the circle.

### Alternate proof of the converse using geometry

Given a right triangle ABC with hypotenuse AC, construct a circle Ω whose diameter is AC. Let O be the center of Ω. Let D be the intersection of Ω and the ray OB. By Thales's theorem, ∠ADC is right. But then D must equal B. (If D lies inside ∆ABC, ∠ADC would be obtuse, and if D lies outside ∆ABC, ∠ADC 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 of their directional vectors 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 AC 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 AC 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.

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 AC 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 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. [6] 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.

## Notes

1. Heath, Thomas L. (1956). The thirteen books of Euclid's elements. New York, NY [u.a.]: Dover Publ. p.  61. ISBN   0486600890.
2. Allen, G. Donald (2000). "Thales of Miletus" (PDF). Retrieved 2012-02-12.
3. Patronis, T.; Patsopoulos, D. The Theorem of Thales: A Study of the naming of theorems in school Geometry textbooks. Patras University . Retrieved 2012-02-12.
4. de Laet, Siegfried J. (1996). History of Humanity: Scientific and Cultural Development. UNESCO, Volume 3, p. 14. ISBN   92-3-102812-X
5. Boyer, Carl B. and Merzbach, Uta C. (2010). A History of Mathematics. John Wiley and Sons, Chapter IV. ISBN   0-470-63056-6
6. Resources for Teaching Mathematics: 14–16 Colin Foster

## Related Research Articles

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 constant. The distance between any point of the circle and the centre is called the radius. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted.

A quadrilateral is a polygon in Euclidean plane geometry with four edges (sides) and four vertices (corners). Other names for quadrilateral include quadrangle and tetragon. A quadrilateral with vertices , , and is sometimes denoted as .

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 .

A right triangle or right-angled triangle (British), or more formally an orthogonal triangle, is a triangle in which one angle is a right angle. The relation between the sides and angles of the right angled is the basis for trigonometry.

In geometry, a solid angle is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The point from which the object is viewed is called the apex of the solid angle, and the object is said to subtend its solid angle from that point.

In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of a triangle to the sines of its angles. According to the law,

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 equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.

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 equals the sum of the squares of the lengths of the other two sides. For example, if one of the other sides has a length of 3 and the other has a length of 4, then their squares add up to 25. The length of the hypotenuse is the square root of 25, that is, 5.

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 between the sine and cosine functions.

In plane geometry, Morley's trisector theorem states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, called the first Morley triangle or simply the Morley triangle. The theorem was discovered in 1899 by Anglo-American mathematician Frank Morley. It has various generalizations; in particular, if all of the trisectors are intersected, one obtains four other equilateral triangles.

In Euclidean geometry, Ptoemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral. The theorem is named after the Greek astronomer and mathematician Ptolemy. Ptolemy used the theorem as an aid to creating his table of chords, a trigonometric table that he applied to astronomy.

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.

In geometry, Apollonius's theorem is a theorem relating the length of a median of a triangle to the lengths of its sides. It states that "the sum of the squares of any two sides of any triangle equals twice the square on half the third side, together with twice the square on the median bisecting the third side".

In mathematics, the sine is a trigonometric function of an angle. The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle, to the length of the longest side of the triangle. For an angle , the sine function is denoted simply as .

The main trigonometric identities between trigonometric functions are proved, using mainly the geometry of the right triangle. For greater and negative angles, see Trigonometric functions.

In trigonometry, the law of cosines 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

In mathematics, the Pythagorean theorem, or Pythagoras's theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides. This theorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:

In geometry, a fat object is an object in two or more dimensions, whose lengths in the different dimensions are similar. For example, a square is fat because its length and width are identical. A 2-by-1 rectangle is thinner than a square, but it is fat relative to a 10-by-1 rectangle. Similarly, a circle is fatter than a 1-by-10 ellipse and an equilateral triangle is fatter than a very obtuse triangle.