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Steiner's problem

Source: 2022 Viet Nam math olympiad for high school students D1 P5

March 20, 2023
geometry

Problem Statement

Given a convex quadrilateral MNPQMNPQ. Assume that there exists 2 points U,VU, V inside MNPQMNPQ satifying:∠MUN=∠MUV=∠NUV=∠QVU=∠PVU=∠PVQ\angle MUN = \angle MUV = \angle NUV = \angle QVU = \angle PVU = \angle PVQConsider another 2 points X,YX, Y in the plane. Prove that the sumXM+XN+XY+YP+YQXM + XN + XY + YP + YQget its minimum value iff X≡U,Y≡VX\equiv U, Y\equiv V.
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