IMO 1979 SL 1

Origin: BEL

Problem

Prove that in the Euclidean plane every regular polygon having an even number of sides can be dissected into lozenges. (A lozenge is a quadrilateral whose four sides are all of equal length).

Solution

We prove more generally, by induction on n, that any 2n-gon with equal edges and opposite edges parallel to each other can be dissected. For n = 2 the only possible such 2n-gon is a single lozenge, so our theo- rem holds in this case. We will now show that it holds for general n. Assume by induction that it holds for n −1. Let A1A2 . . . A2n be an arbitrary 2n-gon with equal edges and opposite edges parallel to each other. Then we can construct points Bi for i = 3, 4, . . ., n such that −−−\to AiBi = −−−\to A2A1 = −−−−−−−\to An+1An+2. We set B2 = A2n+1 = A1 and Bn+1 = An+2. It follows that AiBiBi+1Ai+1 for i = 2, 3, 4, . . ., n are all lozenges. It also follows that BiBi+1 for i = 2, 3, 4, . . ., n are equal to the edges of A1A2 . . . A2n and parallel to AiAi+1 and hence to An+iAn+i+1. Thus B2 . . . Bn+1An+3 . . . A2n is a 2(n −1)-gon with equal edges and opposite sides parallel and hence, by the induction hypothesis, can be dissected into lozenges. We have thus provided a dissection for A1A2 . . . A2n. This completes the proof.