Question 13
Consider a triangle $ABC$ and a point $P$ in its interior. Let the reflections of $P$ in the sides $\overline{BC}$, $\overline{CA}$ and $\overline{AB}$ be denoted by $A_1$, $B_1$ and $C_1$. Further, let $\Omega$ be the circumcircle of triangle $A_1B_1C_1$, and finally let $A_2$, $B_2$ and $C_2$ be the second intersection points of the lines $A_1P$, $B_1P$ and $C_1P$ with $\Omega$. Prove that the three lines $AA_2$, $BB_2$ and $CC_2$ meet at a point on $\Omega$.