Question 6
Let $ABC$ be an isosceles triangle with $AB=AC$, and let $D$ be a point of the segment $BC$ such that $BD\neq CD$. Let $P$ and $Q$ be the perpendicular projections of $D$ onto the lines $AB$ and $AC$, respectively. Finally, let $E$ be the intersection other than $A$ of the circumcircles of $ABC$ and $APQ$.
Prove that if the lines $EP$, $AC$, and the perpendicular bisector of $PQ$ are concurrent, then $ABC$ is right-angled at $A$.
Prove that if the lines $EP$, $AC$, and the perpendicular bisector of $PQ$ are concurrent, then $ABC$ is right-angled at $A$.