IMO 1989 LL GBR25

Origin: GBR

Problem

Let ABC be a triangle. Prove that there is a unique point U in the plane of ABC such that there exist real numbers \lambda, µ, \nu, \kappa, not all zero, such that \lambdaPL2 + µPM 2 + \nuPN 2 −\kappaUP 2 is constant for all points P of the plane, where L, M, N are the feet of the perpendiculars from P to BC, CA, AB respectively.