Question 16
Let $ABC$ be a triangle. Find all points $P$ on segment $BC$ satisfying the following property: If $X$ and $Y$ are the intersections of line $PA$ with the common external tangent lines of the circumcircles of triangles $PAB$ and $PAC$, then
$$
\left( \frac{PA}{XY} \right)^2 + \frac{PB \cdot PC}{AB \cdot AC} = 1.
$$
(This problem was suggested by Titu Andreescu and Cosmin Pohoata.)
$$
\left( \frac{PA}{XY} \right)^2 + \frac{PB \cdot PC}{AB \cdot AC} = 1.
$$
(This problem was suggested by Titu Andreescu and Cosmin Pohoata.)