Question 18
Let a triangle $A B C$ with circumcircle $\Omega$ be given, together with points $A_{1}, B_{1}$ and $C_{1}$ on the sides $\overline{B C}, \overline{C A}$ and $\overline{A B}$ such that the three lines $A A_{1}, B B_{1}$ and $C C_{1}$ have a point $P$ in common.
Prove that at most two of the three reflections of $P$ in the points $A_{1}$, $B_{1}$ and $C_{1}$ (point reflections) lie outside $\Omega$.
Prove that at most two of the three reflections of $P$ in the points $A_{1}$, $B_{1}$ and $C_{1}$ (point reflections) lie outside $\Omega$.