Question 150
Given a circle $\Gamma$ and three fixed points $A, B, C$ on $\Gamma$, and also a real number $\lambda$ with $0 \lt \lambda \lt 1$. Let $P$ be a moving point on $\Gamma$ different from $A, B, C$, and let $M$ be the point on segment $CP$ such that $CM = \lambda \cdot CP$. Let $Q$ be the second intersection point of the circumcircles of triangles $AMP$ and $BMC$. Prove that as $P$ varies, $Q$ lies on a fixed circle.