APMO 2025 Problem 1

Let $A B C$ be an acute triangle inscribed in a circle $\Gamma$ . Let $A_{1}$ be the orthogonal projection of $A$ onto $B C$ so that $A A_{1}$ is an altitude. Let $B_{1}$ and $C_{1}$ be the orthogonal projections of $A_{1}$ onto $A B$ and $A C$ , respectively. Point $P$ is such that quadrilateral $A B_{1}P C_{1}$ is convex and has the same area as triangle $A B C$ . Is it possible that $P$ strictly lies in the interior of circle $\Gamma$ ? Justify your answer.