Question 85
Let $P$ be a point inside triangle $ABC$. Let $AP$ meet $BC$ at $A_{1}$, let $BP$ meet $CA$ at $B_{1}$, and let $CP$ meet $AB$ at $C_{1}$. Let $A_{2}$ be the point such that $A_{1}$ is the midpoint of $PA_{2}$, let $B_{2}$ be the point such that $B_{1}$ is the midpoint of $PB_{2}$, and let $C_{2}$ be the point such that $C_{1}$ is the midpoint of $PC_{2}$. Prove that points $A_{2}, B_{2}$, and $C_{2}$ cannot all lie strictly inside the circumcircle of triangle $ABC$.