Here's a way. Maybe someone has another. We'll see.
Draw the segment from M to the origin (0, 0). Find its distance using the distance formula.
You should find it to be
Find the coordinates of the midpoint M on the hypotenuse using the midpoint formula.
You should come up with
Now find the distances from M to the vertices on the hypotenuse and you'll see they're the same as the distance from M to the origin.