Thales' Theorem - Proof

Proof

We use the following facts: the sum of the angles in a triangle is equal to two right angles (180°) and the base angles of an isosceles triangle are equal.

Provided AC is a diameter, angle at B is constant right (90°).
Figure for the proof.

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 BAO = ABO.

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 two right angles, we have

then

or simply

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