IMO 1986 SL 16

Origin: ISR

Problem

Let A, B be adjacent vertices of a regular n-gon in the plane and let O be its center. Now let the triangle ABO glide around the polygon in such a way that the points A and B move along the whole circumference of the polygon. Describe the figure traced by the vertex O.

Solution

Let Z be the center of the polygon.

Suppose that at some moment we have A \in Pi−1Pi and B \in PiPi+1, where Pi−1, Pi, Pi+1 are ad- jacent vertices of the polygon. Since \angleAOB

180◦−\anglePi−1PiPi+1, the quadrilateral APiBO is cyclic. Hence \angleAPiO = \angleABO = \angleAPiZ, which means that O \inPiZ. Zi A B O Z Pi−1 Pi Pi+1 Moreover, from OPi = 2r sin \anglePiAO, where r is the radius of circle APiBO, we obtain that ZPi \leqOPi \leqZPi/cos(\pi/n). Thus O traces a segment ZZi as A and B move along Pi−1Pi and PiPi+1 respectively, where Zi is a point on the ray PiZ with PiZi cos(\pi/n) = PiZ. When A, B move along the whole circumference of the polygon, O traces an asterisk consisting of n segments of equal length emanating from Z and pointing away from the vertices.