Question 152
Acute triangle $A B C$ has $\angle B A C \lt 45^{\circ}$. Point $D$ lies in the interior of triangle $A B C$ so that $B D = C D$ and $\angle B D C = 4 \angle B A C$. Point $E$ is the reflection of $C$ across line $A B$, and point $F$ is the reflection of $B$ across line $A C$. Prove that lines $A D$ and $E F$ are perpendicular.