Construct squares on the legs of a triangle. Then the lines that join the midpoint of the base with the centers of the square are congruent and perpendicular.

Drag A

Compare the velocities of H and M.

BA is equal to sqrt of 2 times BH, rotated 45^o. Therefore the velocity of A is equal to sqrt 2 times the velocity of H, rotated 45^o. Same way for A and M (rotated 45^o the other way).
Therefore the velocity of H is the same as the velocity of M, rotated 90^o.
Therefore DH = DM rotated 90^o plus a constant. To see that the constant is zero, let A coincide with D.