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Two equations hold then two segments have equal length

Source: Turkey Junior MO 2014 Problem 4

November 16, 2014
geometrycircumcircletrapezoidgeometry proposed

Problem Statement

ABCABC is an acute triangle with orthocenter HH. Points DD and EE lie on segment BCBC. Circumcircle of △BHC\triangle BHC instersects with segments ADAD,AEAE at PP and QQ, respectively. Prove that if BD2+CD2=2DP⋅DABD^2+CD^2=2DP\cdot DA and BE2+CE2=2EQ⋅EABE^2+CE^2=2EQ\cdot EA, then BP=CQBP=CQ.
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