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Sum of distances to sides => perpendicularity

Source: Bundeswettbewerb Mathematik 2015 - Round 2 - #4

September 4, 2015
geometryincentercircumcircledistanceperpendicular

Problem Statement

Let ABCABC be a triangle, such that its incenter II and circumcenter UU are distinct. For all points XX in the interior of the triangle let d(X)d(X) be the sum of distances from XX to the three (possibly extended) sides of the triangle. Prove: If two distinct points P,QP,Q in the interior of the triangle ABCABC satisfy d(P)=d(Q)d(P)=d(Q), then PQPQ is perpendicular to UIUI.
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