Question 83
Circles $\Gamma_{1}$ and $\Gamma_{2}$ intersect into $P$ and $Q$. Let $A$ be a point on $\Gamma_{1}$ not equal to $P$ or $Q$. The lines $A P$ and $A Q$ intersect $\Gamma_{2}$ again into $B$ and $C$, respectively.
Prove that the elevation line from $A$ in triangle $A B C$ passes through a point that is independent of the choice of $A$.
Prove that the elevation line from $A$ in triangle $A B C$ passes through a point that is independent of the choice of $A$.