KöMaL 2014 A614

In the triangle $A_1A_2A_3$, denote by $k_i$ the excircle opposite to $A_i$, and let $P_i$ be the point of $k_i$ for which the circle $A_{i+1}A_{i+2}P_i$ is tangent to $k_i$. ($i=1,2,3$; the indices are considered modulo $3$.) Show that the line segments $A_1P_1$, $A_2P_2$ and $A_3P_3$ are concurrent.