Consider a convex quadrilateral with vertices at $a, b, c$ and $d$ and on each side draw a square lying outside the given quadrilateral, as in the picture below. Let $p, q, r$ and $s$ be the centers of those squares:
a) Find expressions for $p, q, r$ and $s$ in terms of $a, b, c$ and $d$.
b) Prove that the line segment between $p$ and $r$ is perpendicular and equal in length to the line segment between $q$ and $s$.
I managed to do part A, via finding a diagonal and then the midpoint. For part (b), I found a coord-bash algebraic solution, via assigning a lot of variables. However, is there a geometric solution?