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Passing through midpoint

Source: 2022 China Second Round A1

December 22, 2022
geometry

Problem Statement

In acute triangle △ABC\triangle ABC, HH is the orthocenter, BDBD,CECE are altitudes. MM is the midpoint of BCBC. PP,QQ are on segment BMBM,DEDE, respectively. RR is on segment PQPQ such that BPEQ=CPDQ=PRQR\frac{BP}{EQ}=\frac{CP}{DQ}=\frac{PR}{QR}. Suppose LL is the orthocenter of △AHR\triangle AHR, then prove: QMQM passes through the midpoint of RLRL.
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