12th European Girls' Mathematical Olympiad (EGMO 2023)

Problem 6.
Let $ABC$ be a triangle with circumcircle $\Omega$ . Let $S_b$ and $S_c$ respectively denote the midpoints of the arcs $AC$ and $AB$ that do not contain the third vertex. Let $N_a$ denote the midpoint of arc $BAC$ (the arc $BC$ including $A$ ). Let $I$ be the incenter of $ABC$ . Let $\omega_b$ be the circle that is tangent to $AB$ and internally tangent to $\Omega$ at $S_b$ , and let $\omega_c$ be the circle that is tangent to $AC$ and internally tangent to $\Omega$ at $S_c$ . Show that the line $IN_a$ , and the lines through the intersections of $\omega_b$ and $\omega_c$ , meet on $\Omega$ .