95.15 Proofs by dissection of a dodecagon

摘要

Problem: The points T and U lie on the sides QR and RS respectively of square PQRS so that ∠QPT = ∠TPU = ∠UPS = 30° (see Figure 1). Prove that area of triangle PTU = 1/3 area of square PQRS.SUR T PQFIGURE 1 This problem appeared on the Hamilton paper of the 2009 Intermediate Mathematical Olympiad, set by the United Kingdom Mathematics Trust. As candidates demonstrated, there are several different methods of proof, for example, using trigonometry. However, since the result only concerns areas, the challenge is to find a proof using a dissection method.