Thales was a Greek mathematician who lived in 654-625 BC. He was considered by many as the first mathematician. One of his famous achievements was devising the first formal geometric proof. The theorem he proved is what is now known as Thales’ Theorem.

**The Theorem**

Thales’ Theorem states that if , , and are points on the circle, and is a diameter, then angle a right angle.You should explore this by drawing circles and different sizes and drawing the triangle. You may also want to use a dynamic software.

**The Proof**

The proof of Thales’ Theorem is quite easy. First, we draw line segment to form two isosceles triangles and . Now, the base of the isosceles triangles are congruent as indicated by and in the second figure. The sum of the interior angles of a triangle is . So,

.

Simplifying, we have

Dividing both sides by 2, we have

But is the measure of angle .

Therefore, the measure of angle is . And we are done.

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