Problem

Source: RMO 2018 P6

Tags: geometry, incenter



Let $ABC$ be an acute-angled triangle with $AB<AC$. Let $I$ be the incentre of triangle $ABC$, and let $D,E,F$ be the points where the incircle touches the sides $BC,CA,AB,$ respectively. Let $BI,CI$ meet the line $EF$ at $Y,X$ respectively. Further assume that both $X$ and $Y$ are outside the triangle $ABC$. Prove that $\text{(i)}$ $B,C,Y,X$ are concyclic. $\text{(ii)}$ $I$ is also the incentre of triangle $DYX$.