APMO 2025 - Asian Pacific Mathematics Olympiad Question 1

Let $ABC$ be an acute triangle inscribed in a circle $\Gamma$. Let $A_1$ be the orthogonal projection of $A$ onto $BC$, so $AA_1$ is an altitude. Let $B_1,C_1$ be the orthogonal projections of $A_1$ onto $AB,AC$, respectively. Point $P$ is such that quadrilateral $AB_1PC_1$ is convex and has the same area as triangle $ABC$. Is it possible that $P$ lies strictly inside $\Gamma$? Justify your answer.

AI-generated diagram of the given triangle and projections: