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ABCD is parallelogram if h(A)=h(B) =h(C)=h(D), sum of distances from 2 sides

Source: 1967 Hungary - Kürschák Competition p3

October 11, 2022
geometryparallelogram

Problem Statement

For a vertex XX of a quadrilateral, let h(X)h(X) be the sum of the distances from XX to the two sides not containing XX. Show that if a convex quadrilateral ABCDABCD satisfies h(A)=h(B)=h(C)=h(D)h(A) = h(B) = h(C) = h(D), then it must be a parallelogram.
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