Question 84
Let $H$ and $\Gamma$ be the orthocenter and the circumcircle of an acute triangle $ABC$, respectively, and let $M$ be the midpoint of side $BC$. Take a point $D$ on the minor arc $BC$ of $\Gamma$ such that $\angle BAD = \angle MAC$, and take points $E$ on $\Gamma$ and $F$ on line $BC$ such that $DE$ and $DF$ are perpendicular to $AM$ and $BC$, respectively. Let $N$ be the intersection of lines $HF$ and $AM$, and let $R$ be the reflection of $H$ in $N$.
Prove that $\angle AER + \angle DFR = 180^\circ$.
Prove that $\angle AER + \angle DFR = 180^\circ$.