Question 3
Let $\triangle ABC$ be an acute angled triangle with $AC \gt AB$ and incircle $\omega$. Let $\omega$ touch the sides $BC, CA$, and $AB$ at $D, E$, and $F$ respectively. Let $X$ and $Y$ be points outside $\triangle ABC$ satisfying
$$
\angle BDX = \angle XEA = \angle YDC = \angle AFY = 45^\circ.
$$
Prove that the circumcircles of $\triangle AXY$, $\triangle AEF$ and $\triangle ABC$ meet at a point $Z \neq A$.
$$
\angle BDX = \angle XEA = \angle YDC = \angle AFY = 45^\circ.
$$
Prove that the circumcircles of $\triangle AXY$, $\triangle AEF$ and $\triangle ABC$ meet at a point $Z \neq A$.