Problem

Source: ELMO Shortlist 2024 G3

Tags: geometry, Elmo



Let $ABC$ be a triangle, and let $\omega_1,\omega_2$ be centered at $O_1$, $O_2$ and tangent to line $BC$ at $B$, $C$ respectively. Let line $AB$ intersect $\omega_1$ again at $X$ and let line $AC$ intersect $\omega_2$ again at $Y$. If $Q$ is the other intersection of the circumcircles of triangles $ABC$ and $AXY$, then prove that lines $AQ$, $BC$, and $O_1O_2$ either concur or are all parallel. Advaith Avadhanam