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quadrilateral of orthocenters is a parallelogram

Source: 1998 Argentina OMA L3 p2

June 11, 2020
geometryorthocentertangential quadrilateralparallelogram

Problem Statement

Let a quadrilateral ABCDABCD have an inscribed circle and let K,L,M,NK, L, M, N be the tangency points of the sides AB,BC,CDAB, BC, CD and DADA, respectively. Consider the orthocenters of each of the triangles △AKN,△BLK,△CML\vartriangle AKN, \vartriangle BLK, \vartriangle CML and △DNM\vartriangle DNM. Prove that these four points are the vertices of a parallelogram.
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