IMO 2000 SL G4

Origin: RUS | Category: Geometry

Problem

Let A1A2 . . . An be a convex polygon, n \geq4. Prove that A1A2 . . . An is cyclic if and only if to each vertex Aj one can assign a pair (bj, cj) of real numbers, j = 1, 2, . . .n, such that AiAj = bjci −bicj for all i, j with 1 \leqi \leqj \leqn.

Solution

First, suppose that there are numbers (bi, ci) assigned to the vertices of the polygon such that AiAj = bjci −bicj for all i, j with 1 \leqi \leqj \leqn. (1) In order to show that the polygon is cyclic, it is enough to prove that A1, A2, A3, Ai lie on a circle for each i, 4 \leqi \leqn, or equivalently, by Ptolemy’s theorem, that A1A2 \cdot A3Ai + A2A3 \cdot AiA1 = A1A3 \cdot A2Ai. But this is straightforward with regard to (1). Now suppose that A1A2 . . . An is a cyclic quadrilateral. By Ptolemy’s the- orem we have AiAj = A2Aj \cdot A1Ai A1A2 −A2Ai \cdot A1Aj A1A2 for all i, j. This suggests taking b1 = −A1A2, bi = A2Ai for i \geq2 and ci = A1Ai A1A2 for all i. Indeed, using Ptolemy’s theorem, one easily verifies (1).