Problem

Source: Romanian JBTST VI 2007, problem 3

Tags: geometry, geometric transformation, reflection, trigonometry, angle bisector, power of a point, radical axis



Let $ABC$ be a right triangle with $A = 90^{\circ}$ and $D \in (AC)$. Denote by $E$ the reflection of $A$ in the line $BD$ and $F$ the intersection point of $CE$ with the perpendicular in $D$ to $BC$. Prove that $AF, DE$ and $BC$ are concurrent.