1969 IMO Problems/Problem 4

A semicircular arc $\gamma$ is drawn with $AB$ as diameter. $C$ is a point on $\gamma$ other than $A$ and $B$ , and $D$ is the foot of the perpendicular from $C$ to $AB$ . We consider three circles, $\gamma_1, \gamma_2, \gamma_3$ , all tangent to the line $AB$ . Of these, $\gamma_1$ is inscribed in $\triangle ABC$ , while $\gamma_2$ and $\gamma_3$ are both tangent to $CD$ and $\gamma$ , one on each side of $CD$ . Prove that $\gamma_1, \gamma_2$ , and $\gamma_3$ have a second tangent in common.

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1969 IMO Problems/Problem 4