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Four midpoints are cyclic

Source: Czech-Polish-Slovak Match 2019 P6

July 15, 2019
geometry

Problem Statement

Let ABCABC be an acute triangle with AB<ACAB<AC and ∠BAC=60∘\angle BAC=60^{\circ}. Denote its altitudes by AD,BE,CFAD,BE,CF and its orthocenter by HH. Let K,L,MK,L,M be the midpoints of sides BC,CA,ABBC,CA,AB, respectively. Prove that the midpoints of segments AH,DK,EL,FMAH, DK, EL, FM lie on a single circle.
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