Right triangle — the median to the hypotenuse equals half the hypotenuse

Dependencies: Parallelogram tests (F.04), Rectangle tests (F.07), Rectangle: equal diagonals (F.06).

Statement

Let $\triangle ABC$ have $\angle C = 90^\circ$, and let $M$ be the midpoint of the hypotenuse $AB$. Then the median to the hypotenuse has length exactly half the hypotenuse, and $M$ is equidistant from the three vertices:

In other words, the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices — another way of stating that it is the centre of the circumscribed circle, with the hypotenuse as the diameter.