KöMaL 2005 A368

The interior bisector of the angle $A$ of the triangle ABC cuts the incircle $c$ at two points; the one closer to $A$ is denoted by $O_A$. Define the points $O_B$ and $O_C$ similarly on the internal bisectors of the angles $B$ and $C$, respectively. The circle $c_A$ is drawn about $O_A$ and it is touching the sides $AB$ and $CA$. Similarly, $c_B$ is about $O_B$ and it is touching the sides $BC$ and $AB$, finally the circle $c_C$ is about $O_C$ and it is touching the sides $CA$ and $BC$. Taken any two of the circles $c_A$, $c_B$ and $c_C$ consider their common external tangents different from the corresponding sides of the triangle, respectively. Prove that these three tangents are concurrent.