Why is the midpoint of a right angled triangle side always on the same height?

Question

it's been a great while since I've had basic geometry, and this seems to be elude me. Here are two images of a triangle $DMC$: How can I prove that the midpoint of $DC$ (point $E$) will always be in the same height (the pink line) and that this height is exactly half the length of $MC$? I've tried using some triangle similarities but I was somehow unable to prove this universally.

Answer 1

The parallel red and green lines create similar triangles. There's no need for the triangles to be right triangles.