Question 3
Let three circles $\Gamma_1,\Gamma_2,\Gamma_3$, which are non-overlapping and mutually external, be given in the plane. For each point $P$ in the plane, outside the three circles, construct six points $A_1,B_1,A_2,B_2,A_3,B_3$ as follows: for each $i=1,2,3$, $A_i,B_i$ are distinct points on $\Gamma_i$ such that $PA_i$ and $PB_i$ are both tangents to $\Gamma_i$. Call $P$ exceptional if the three lines $A_1B_1,A_2B_2,A_3B_3$ are concurrent. Show that every exceptional point of the plane, if any exists, lies on the same circle.
Illustrative configuration (newly drawn):
Illustrative configuration (newly drawn):