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

If $AB$ is a diameter of a circle and $C$ is any other point on the circle, then the inscribed angle is a right angle:

Strategy: this is a special case of the inscribed angle theorem. The diameter $AB$ subtends a straight central angle of $180°$, so the inscribed angle $\angle ACB$ is half of that, $90°$. The converse follows by running the same argument backwards.

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:

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 $AB$.

Proof: Every triangle has a unique circumscribed circle. Let $O$ be its centre, so $|OA|=|OB|=|OC|$.

Apply the inscribed angle theorem on this circle:

A straight central angle means $A$, $O$, $B$ are collinear, so $O$ lies on segment $AB$ — hence $AB$ is a diameter, and $C$ lies on the circle with diameter $AB$.

Corollary: The hypotenuse of a right triangle is the diameter of its circumscribed circle.

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

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. ∎