MathDB

6

Part of 2019 Czech-Austrian-Polish-Slovak Match

Problems(1)

Four midpoints are cyclic

Source: Czech-Polish-Slovak Match 2019 P6

7/15/2019
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.
geometry