Question 42
We are given an acute triangle $ABC$. Point $D$ lies in the halfplane $AB$ containing $C$ and satisfies $DB \perp AB$ and $\angle ADB = 45^\circ + \frac{1}{2}\angle ACB$. Similarly, $E$ lies in the halfplane $AC$ containing $B$ and satisfies $AC \perp EC$ and $\angle AEC = 45^\circ + \frac{1}{2}\angle ABC$. Let $F$ be the reflection of $A$ in the midpoint of arc $BAC$ (containing point $A$). Prove that points $A, D, E, F$ are concyclic. (Patrik Bak, Slovakia)