IMO 1992 SL 20

Origin: FRA

Problem

In the plane, let there be given a circle C, a line l tangent to C, and a point M on l. Find the locus of points P that have the following property: There exist two points Q and R on l such that M is the midpoint of QR and C is the incircle of PQR.

Solution

Denote by U the point of tangency of the circle C and the line l. Let X and U ′ be the points symmetric to U with respect to S and M respectively; these points do not depend on the choice of P. Also, let C′ be the excircle of \trianglePQR corresponding to P, S′ the center of C′, and W, W ′ the points of tangency of C and C′ with the line PQ respectively. Ob- viously, \triangleWSP ∼\triangleW ′S′P. Since SX \parallelS′U ′ and SX : S′U ′

SW : S′W ′ = SP : S′P, we de- duce that ∆SXP ∼∆S′U ′P, and consequently that P lies on the line XU ′. On the other hand, it is easy to show that each point P of the ray U ′X over X satisfies the required condition. Thus the desired locus is the extension of U ′X over X. U U ′ M P S′ S W Q R W ′ X