KöMaL 2010 B4287

The centres of the escribed circles of triangle ABC are $O_1$, $O_2$ and $O_3$. $P$ is an interior point of the triangle, different from the centre of the inscribed circle. The feet of the perpendiculars drawn from $P$ to the angle bisectors are $M_1$, $M_2$ and $M_3$. Prove that the triangles $O_1 O_2 O_3$ and $M_1 M_2 M_3$ are similar.