KöMaL 2025 A920

Given a non-isosceles triangle $ABC$, let its circumcircle be denoted by $\Omega$. Let $\omega_A$ be the mixtilinear incircle corresponding to vertex $A$ (that is, the circle tangent to sides $AB$, $AC$, and internally tangent to $\Omega$). Define $\omega_B$ and $\omega_C$ analogously. Let the points where $\omega_A,\omega_B,\omega_C$ touch $\Omega$ be denoted by $T_A, T_B, T_C$, respectively. Show that the Monge line of the circles $\omega_A,\omega_B,\omega_C$ (that is, the line passing through the pairwise external homothety centers of the three circles) coincides with the Pascal line of the hexagon $AT_CBT_ACT_B$ (that is, the line passing through the points $AT_C \cap T_AC$, $T_CB \cap CT_B$, $BT_A \cap T_BA$).