KöMaL 2023 A862

Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\omega$. Let $F_A$, $F_B$, $F_C$ and $F_D$ be the midpoints of arcs $AB$, $BC$, $CD$ and $DA$ of $\omega$. Let $I_A$, $I_B$, $I_C$ and $I_D$ be the incenters of triangles $DAB$, $ABC$, $BCD$ and $CDA$, respectively. Let $\omega_A$ denote the circle that is tangent to $\omega$ at $F_A$ and also tangent to line segment $CD$. Similarly, let $\omega_C$ denote the circle that is tangent to $\omega$ at $F_C$ and tangent to line segment $AB$. Finally, let $T_B$ denote the second intersection of $\omega$ and circle $F_BI_BI_C$ different from $F_B$, and let $T_D$ denote the second intersection of $\omega$ and circle $F_DI_DI_A$. Prove that the radical axis of circles $\omega_A$ and $\omega_C$ passes through points $T_B$ and $T_D$.