The angle A of the convex quadrilateral ABCD is 90 °, and the vertex C is removed from lines AB and AD

The angle A of the convex quadrilateral ABCD is 90 °, and the vertex C is removed from lines AB and AD by distances equal to the valleys of the segments AB and AD, respectively. Prove that the diagonals of ABCD are perpendicular.

Let us prove that the triangles AВD and AСН are equal.
Since the quadrangle AKSN is rectangular, then AH = KC = AB. The ABP leg of the AED triangle is equal to the CH leg of the AСН triangle, then the AED triangle is equal to the AСН triangle along two legs – the first sign of the triangle equality. Angle 1 of triangle AKС is equal to angle 1 of triangle ACН as criss-crossing angles at the intersection of parallel lines AB and CH secant AC. Then the angle AСD = ВDA.
The sum of angles 1 and 2 is 900, then, in the triangle AOD, the angle AOD = 180 – OAD – ODA = 180 – (OAD + ODA) = 180 – 90 = 900. Then the ВD is perpendicular to the AС, which was required to be proved.