Question 101
Two circles $\gamma$ and $\gamma'$ cross one another at points $A$ and $B$. The tangent to $\gamma'$ at $A$ meets $\gamma$ again at $C$, the tangent to $\gamma$ at $A$ meets $\gamma'$ again at $C'$, and the line $CC'$ separates the points $A$ and $B$. Let $\Gamma$ be the circle externally tangent to $\gamma$, externally tangent to $\gamma'$, tangent to the line $CC'$, and lying on the same side of $CC'$ as $B$. Show that the circles $\gamma$ and $\gamma'$ intercept equal segments on one of the tangents to $\Gamma$ through $A$.