Let's take ABCD a convex quadrilateral, such that the diagonals AC and BD are perpendicular, and the angles ABC and CDA are equal.
What is the easiest way to prove that the triangles ABC and CDA are congruent?
My solution builds the point E, that is the symmetric of D with respect to AC, and then A,E,B,C must be on the same circle, but this is possible only if E=B, so the triangles must be congruent.