Question 35
Let three circles $\Gamma_1,\Gamma_2,\Gamma_3$, with centres $A_1,A_2,A_3$ and radii $r_1,r_2,r_3$ respectively, be mutually tangent to each other externally. Suppose that the tangent to the circumcircle of the triangle $A_1A_2A_3$ at $A_3$ and the two external common tangents of $\Gamma_1$ and $\Gamma_2$ meet at a common point $P$, as shown in the figure below. Given that $r_1=18$ cm, $r_2=8$ cm and $r_3=k$ cm, find the value of $k$.
