Let ABCD be a square and Z the midpoint of BC. From vertex A, draw a perpendicular to line DZ with foot E. Draw segment EC. Find ∠ZEC.

Context:

I was studying parallelograms and their properties, especially special cases like squares. Working with square ABCD, I took Z as the midpoint of side BC and drew line DZ. From vertex A, I drew a perpendicular to DZ with foot at point E, then drew segment EC. I tried to find this angle synthetically and wondered if there was some symmetry at play. So I extended AE until it met side CD at point F, and by analyzing the quadrilateral EFCZ that formed, I concluded that the angle equals 45°. I'm interested in seeing other approaches - perhaps using geometric transformations (rotations, reflections), coordinate geometry, or another synthetic proof without auxiliary lines. I'm also curious if there's any generalization of this result.