A property of the midpoint of the hypotenuse in a right triangle

Any right triangle can be inscribed into a circle with center $C$ where the hypotenuse $OB$ is the diameter. Now it is straightforward to deduce that $CA$ is a radius, so $CO=CB=CA=r$.

This is a result of elementary geometry, in fact it is possible demonstrate that in any circle the angle at the centre is twice the corresponding angle at the circumference: $\angle OCB=2\angle OAB$: Euclid's inscribed angle theorem.

Now, if the angle $\angle OCB=180$, then $\angle OAB=90$, $OB$ is a diameter and $\triangle OAB$ is a right triangle.