Question 47
(NET 2) (IMO 4) Let $A$ and $B$ be points on the circle $\gamma$. A point $C$, different from $A$ and $B$, is on the circle $\gamma$. Let $D$ be the projection of the point $C$ onto the line $AB$. Consider three other circles $\gamma_1$, $\gamma_2$, and $\gamma_3$ with the common tangent $AB$: $\gamma_1$ inscribed in the triangle $ABC$, and $\gamma_2$ and $\gamma_3$ tangent to both (the segment) $CD$ and $\gamma$. Prove that $\gamma_1$, $\gamma_2$, and $\gamma_3$ have two common tangents.