Question 45
Let $ABC$ be a triangle with $\angle A \lt 60^{\circ}$. Let $X$ and $Y$ be the points on the sides $AB$ and $AC$, respectively, such that $CA + AX = CB + BX$ and $BA + AY = BC + CY$. Let $P$ be the point in the plane such that the lines $PX$ and $PY$ are perpendicular to $AB$ and $AC$, respectively. Prove that $\angle BPC \lt 120^{\circ}$.