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An easy geometry in Taiwan TST

Source: 2022 Taiwan TST Round 3 Independent Study 1-G

April 27, 2022
geometrycircumcirclegeometric transformationreflection

Problem Statement

Let ABCABC be an acute triangle with orthocenter HH and circumcircle Ω\Omega. Let MM be the midpoint of side BCBC. Point DD is chosen from the minor arc BCBC on Γ\Gamma such that ∠BAD=∠MAC\angle BAD = \angle MAC. Let EE be a point on Γ\Gamma such that DEDE is perpendicular to AMAM, and FF be a point on line BCBC such that DFDF is perpendicular to BCBC. Lines HFHF and AMAM intersect at point NN, and point RR is the reflection point of HH with respect to NN.
Prove that ∠AER+∠DFR=180∘\angle AER + \angle DFR = 180^\circ.
Proposed by Li4.
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