Thales Theorem

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 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 valid, which states that a Right Triangle 's hypotenuse is a diameter of its Circumcircle . 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. Proof of the converse The proof utilises the fact that directional Vector s of two lines form right angles if and only if the Dot Product is zero. 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 the Dot Product of AB and BC is
	:A	B

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