this is question 42 in the red book of mathematical problems by k. s. williams and k. hardy.
let abcd be a convex quadrilateral. let p be the point outside abcd such that $|ap| = |pb|$ and $\angle apb = 90^\circ.$ the points $q, r, s$ are similarly defined. prove that the lines $pr$ and $qs$ are equal length and perpendicular.
you can make a physical model by cutting four isoscles right triangles of various lengths from a rectangular card. spread them out on a table with the right angles in the exterior. the result sated in the problem can be seen to hold.
the solution in the book use complex numbers and i can solve using the cosine rule. i am wondering if there is a more "geometric" solution with as little as computation as possible.