Right angle ⇒ C lies on the circle with the hypotenuse as diameter (Thales converse)

Depends on: Thales: angle on a diameter = 90°, Linear pair sums to 180°, Through a point off a line, exactly one parallel exists (Playfair).

Statement

Let $\triangle ABC$ have $\angle ACB = 90^\circ$. Then $C$ lies on the circle $\odot O$ with the hypotenuse $AB$ as diameter, where $O$ is the midpoint of $AB$; consequently

In other words, "$\angle C = 90^\circ$" and "$C$ lies on the circle with $AB$ as diameter" are necessary and sufficient for each other — this is the converse of the Thales circle theorem.