Question 92
Let $X Y$ be a chord of a circle $\Omega$, with center $O$, which is not a diameter. Let $P, Q$ be two distinct points inside the segment $X Y$, where $Q$ lies between $P$ and $X$. Let $\ell$ be the perpendicular line dropped from $P$ to the diameter which passes through $Q$. Let $M$ be the intersection point of $\ell$ and $\Omega$ which is closer to $P$. Prove that
$$
M P \cdot X Y \geq 2 \cdot Q X \cdot P Y
$$
$$
M P \cdot X Y \geq 2 \cdot Q X \cdot P Y
$$