KöMaL 2009 A491

In the triangle $A_1A_2A_3$, for each $i=1,2,3$, the excircle, which is tangent to the side $A_{i+1}A_{i+2}$, touches the half lines $A_iA_{i+1}$ and $A_iA_{i+2}$ at $P_i$ and $Q_i$, respectively. (The indices are considered modulo 3, e.g. $A_4=A_1$ and $A_5=A_2$.) The lines $P_iP_{i+1}$ and $Q_iQ_{i+2}$ meet at $R_i$; finally, the lines $P_{i+1}P_{i+2}$ and $Q_{i+1}Q_{i+2}$ meet at $S_i$ ($i=1,2,3$). Prove that the lines $R_1S_1$, $R_2S_2$ and $R_3S_3$ are concurrent.