MathDB
orthocenter wanted

Source: VI Soros Olympiad 1990-00 R1 10.9 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

May 28, 2024
geometryorthocenter

Problem Statement

Given an acute-angled triangle ABCABC, in which PP, MM, NN are the midpoints of the sides ABAB, BCBC, ACAC, respectively. A point HH is taken inside the triangle and perpendiculars HKHK, HSHS, HQHQ are lowered from it to the sides ABAB, BCBC, ACAC, respectively (K∈ABK \in AB, S∈BCS \in BC, Q∈ACQ \in AC). It turned out that MK=MQMK = MQ, NS=NKNS = NK, PS=PQPS=PQ. Prove that HH is the point of intersection of the altitudes of triangle ABCABC.
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