Nemotron-IMO-Bench 2026 G3

Let $\omega_1,\ldots, \omega_4$ be the incircles of the four triangles formed by three vertices of a cyclic quadrilateral $ABCD$, and suppose that their radii are pairwise distinct. For the circles $\omega_i$ and $\omega_j$ (with $1\leq i<j\leq 4$) consider the point $P_{ij}$ where their external common tangents cross. Prove that the points $P_{ij}$ lie on the same line.