Thales' Theorem

A proof that any angle inscribed in a semicircle is a right angle, as a direct corollary of the inscribed angle theorem, together with its converse.

Keywords: Thales' theorem, semicircle, right angle, inscribed angle, diameter

Prerequisites: inscribed-angle-proof · Difficulty: beginner

Thales' Theorem

Theorem: If $AB$ is a diameter of a circle and $C$ is any other point on the circle, then $\angle ACB = 90°$.

Proof Strategy

This is a special case of the inscribed angle theorem. The diameter $AB$ subtends a central angle of $180°$ (a straight angle). By the inscribed angle theorem:

\angle ACB = \tfrac{1}{2} \cdot 180° = 90°
    

Converse

If $\angle ACB = 90°$, then $C$ lies on the circle with diameter $AB$. This is because the inscribed angle must be half the central angle, so the central angle is $180°$, meaning $AB$ is a diameter.

The Proof

Step 1: A Triangle in a Semicircle

Draw a circle with diameter $AB$ and pick any point $C$ on the circle (not at $A$ or $B$). We want to show $\angle ACB = 90°$.

Step 2: Applying the Inscribed Angle Theorem

The diameter $AB$ subtends a straight angle $\angle AOB = 180°$ at the center. By the inscribed angle theorem:

\angle ACB = \tfrac{1}{2} \cdot \angle AOB
      = \tfrac{1}{2} \cdot 180° = 90°
    

Therefore $\angle ACB$ is a right angle!

Step 3: The Converse: Right Angle Implies Semicircle

Converse: If $\angle ACB = 90°$, then $C$ lies on the circle with diameter $AB$.

Why? If $\angle ACB = 90°$, the central angle must be $2 \cdot 90° = 180°$. A central angle of $180°$ means $AB$ is a diameter. So $C$ lies on the circle with diameter $AB$.

This tells us the hypotenuse of a right triangle is the diameter of its circumscribed circle.

Conclusion

Thales' theorem: An angle inscribed in a semicircle is always a right angle:

AB \text{ is a diameter} \implies \angle ACB = 90°
    

Converse: If $\angle ACB = 90°$, then $C$ lies on the circle with diameter $AB$. In other words, the hypotenuse of a right triangle is the diameter of its circumscribed circle.