geometry puzzle

Geometry puzzle (19)

Problem:
Above we see isosceles right angled triangle ABC.
Proof that angle x = LDCE = 45 degrees.

The lengths AD=3, DE=5 and EB=4 show a Pythagoras triple.
So we approach this problem in a more general way:
If AD=a, DE=c and EB=b and also a² + b² = c²
then prove that x = 45.

The proof

Shift lines AD and EB parallel to AB until they meet at point E.
Because of similarity of ΔABC and ΔDEM (equal angles) MD = ME.
We prove that also MC = MD = ME.

Note: in a right angled isosceles triangle with a hypothenuse of a, the other sides are
half the root of a.

M is the center of a circle through points D, E and C.
LDME spans arc DE of 90 degrees.
Angle x also spans arc DE so LDCE = 45.