Problem

Source: 2021 Taiwan TST Round 1 Independent Study 2-G

Tags: geometry, incenter, circumcircle



Let $ABC$ be a triangle with incenter $I$ and circumcircle $\Omega$. A point $X$ on $\Omega$ which is different from $A$ satisfies $AI=XI$. The incircle touches $AC$ and $AB$ at $E, F$, respectively. Let $M_a, M_b, M_c$ be the midpoints of sides $BC, CA, AB$, respectively. Let $T$ be the intersection of the lines $M_bF$ and $M_cE$. Suppose that $AT$ intersects $\Omega$ again at a point $S$. Prove that $X, M_a, S, T$ are concyclic. Proposed by ltf0501 and Li4