IMO 1992 LL IND27

Origin: IND

Problem

Let ABC be an arbitrary scalene triangle. Define \Sigma to be the set of all circles y that have the following properties: (i) y meets each side of \triangleABC in two (possibly coincident) points; (ii) if the points of intersection of y with the sides of the triangle are la- beled by P, Q, R, S, T , U, with the points occurring on the sides in orders B(B, P, Q, C), B(C, R, S, A), B(A, T, U, B), then the following relations of parallelism hold: TS\parallelBC; PU\parallelCA; RQ\parallelAB. (In the lim- iting cases, some of the conditions of parallelism will hold vacuously; e.g., if A lies on the circle y, then T , S both coincide with A and the relation TS\parallelBC holds vacuously.) (a) Under what circumstances is \Sigma nonempty? (b) Assuming that \Sigma is nonempty, show how to construct the locus of centers of the circles in the set \Sigma. (c) Given that the set \Sigma has just one element, deduce the size of the largest angle of \triangleABC. (d) Show how to construct the circles in \Sigma that have, respectively, the largest and the smallest radii.