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R,D,F is collinear

Source: 2021 Korea Winter Program Test2 Day1 #3

February 13, 2021
geometry

Problem Statement

The acute triangle ABCABC satisfies AB‾<BC‾<CA‾\overline {AB}<\overline {BC}<\overline {CA}. Let HH a orthocenter of ABCABC, DD a intersection point of AHAH and BCBC, EE a intersection point of BHBH and ACAC, and MM a midpoint of segment BCBC. A circle with center EE and radius AEAE intersects the segment ACAC at point FF(≠A\neq A), and circumcircle of triangle BFCBFC intersects the segment AMAM at point SS. Let PP(≠D\neq D), QQ(≠F\neq F) a intersection point of circumcircle of triangle ASDASD and DFDF, circumcircle of triangle ASFASF and DFDF respectively. Also, define RR as a intersection point of circumcircles of triangle AHQAHQ and AEPAEP. Prove that RR lies on line DFDF.
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