KöMaL problem A, 2025

In triangle $ABC$, interior point $D$ is chosen such that triangle $BCD$ is equilateral. Let $E$ be the isogonal conjugate of point $D$ with respect to triangle $ABC$. Define point $P$ on the ray $AB$ such that $AP = BE$. Similarly, define point $Q$ on the ray $AC$ such that $AQ = CE$. Prove that line $AD$ bisects segment $PQ$.