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# Thales' theorem

In geometry, Thales' theorem (named after Thales of Miletus) states that if A, B and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle.

## Proof

We use the following facts: the sum of the angles in a triangle is equal to two right angles and that the base angles of an isosceles triangle are equal.

Let O be the center of the circle. Since OA = OB = OC, OAB and OBC are isosceles triangles, and by the equality of the base angles of an isosceles triangle, OBC = OCB and BAO = ABO. Let γ = BAO and δ = OBC.

Since the sum of the angles of a right triangle is equal to two right angles, we have

2γ + γ ′ = 180°

and

2δ + δ ′ = 180°

We also know that

γ ′ + δ ′ = 180°

Adding the first two equations and subtracting the third, we obtain

2γ + γ ′ + 2δ + δ ′ − (γ ′ + δ ′) = 180°

which, after cancelling γ ′ and δ ′, implies that

γ + δ = 90°

Q.E.D

## Converse

The converse of Thales' theorem is also true. It states that if you have a right triangle and construct a circle with the triangle's hypothenuse as diameter, then the third vertex of the triangle will also lie on the circle.

The theorem and its converse can be expressed as follows:

The center of the circumcircle of a triangle lies on one of the triangle's sides if and only if the triangle is a right triangle.

## Generalization

Thales' 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.