KöMaL 2023 A855

In scalene triangle $ABC$ the shortest side is $BC$. Let points $M$ and $N$ be chosen on sides $AB$ and $AC$, respectively, such that $BM=CN=BC$. Let $I$ and $O$ denote the incenter and circumcentre of triangle $ABC$, and let $D$ and $E$ denote the incenter and circumcenter of triangle $AMN$. Prove that lines $IO$ and $DE$ intersect each other on the circumcircle of triangle $ABC$.