Problem 2, 2009

Let \(C\) be a point of the segment \(AB\) other than \(A\) and \(B\), and let \(k_{0}, k_{01}, k_{02}\) be the circles with diameters \(AB, AC\) and \(CB\) respectively. Let \(D\) be a point in which the perpendicular to \(AB\) at \(C\) meets the circle \(k_{0}\).

In the half-plane bounded by the line \(AB\) that contains \(D\), let \(k_{1}\) be a circle touching \(k_{01}\), \(k_{0}\) and the segment \(CD\), and let \(k_{2}\) be a circle touching \(k_{02}\), \(k_{0}\) and the segment \(CD\). Finally, let \(k\) be the circle of smallest radius that contains and touches both \(k_{1}\) and \(k_{2}\).

Prove that the diameter of \(k\) is equal to the length of \(CD\).