MathNet 2023-part-02 — 191 problems (EN–VI) Question 103

Let $ABC$ be a triangle all of whose angles are acute. Let $F$ be the foot of the altitude of $ABC$ from $A$, and let $P$ be a point on the segment $[AF]$. Let $D$ be the point of $(BC)$ such that $(PD)$ is parallel to $(AC)$, and then $X$ the second intersection point of the circumcircle of $ABD$ and the circle with centre $D$ passing through $A$. Similarly, let $E$ be the point of $(BC)$ such that $(PE)$ is parallel to $(AB)$, and then $Y$ the second intersection point of the circumcircle of $ACE$ and the circle with centre $E$ passing through $A$.

Prove that the points $B$, $C$, $X$ and $Y$ are concyclic.