The Cramer-Castillon problem and Urquhart's `most elementary' theorem. (Q960581)

Summary: A very old branch of mathematics, elementary plane geometry, is still full of mysteries. For instance, there is no elegant elementary proof for the ``most'' elementary theorem by Urquhart: ``Let the points \(APQ\), \(ARS\), \(PBS\) and \(QBR\) lie on straight lines, then \(AP + PB = AR + RB\) implies \(AQ + QB = AS + SB\)''. The proof given here uses the Möbius transform which formerly has been invented for another famous problem, the ``Cramer-Castillon problem''. The article arose from an elementary course of lectures on geometry.