2019 IMO Shortlist G4

Let $P$ be a point inside triangle $A B C$. Let $A P$ meet $B C$ at $A_{1}$, let $B P$ meet $C A$ at $B_{1}$, and let $C P$ meet $A B$ at $C_{1}$. Let $A_{2}$ be the point such that $A_{1}$ is the midpoint of $P A_{2}$, let $B_{2}$ be the point such that $B_{1}$ is the midpoint of $P B_{2}$, and let $C_{2}$ be the point such that $C_{1}$ is the midpoint of $P C_{2}$. Prove that points $A_{2}, B_{2}$, and $C_{2}$ cannot all lie strictly inside the circumcircle of triangle $A B C$. (Australia)

令 $P$ 为三角形 $A B C$ 内的一点。让$A P$ 在$A_{1}$ 与$B C$ 相交，让$B P$ 在$B_{1}$ 与$C A$ 相交，并让$C P$ 在$C_{1}$ 与$A B$ 相交。令$A_{2}$ 为点，使得$A_{1}$ 为$P A_{2}$ 的中点，令$B_{2}$ 为点，使得$B_{1}$ 为$P B_{2}$ 的中点，并令$C_{2}$ 为点，使得$C_{1}$ 为$P C_{2}$ 的中点。证明点 $A_{2}、B_{2}$ 和 $C_{2}$ 不能全部严格位于三角形 $A B C$ 的外接圆内。 (澳大利亚)