MathNet 2018-part-05 — 181 problems (EN–VI) Question 31

A triangle $ABC$ has circumcenter $O$ and incenter $I$. The straight lines $AI$, $BI$, $CI$ meet again the circumcircle at $D$, $E$, respectively $F$. Denote $M$, $N$, $P$ the circumcenters of the triangles $IEF$, $IDF$, respectively $IED$. Prove that the straight lines $DM$, $EN$, $FP$ and $OI$ are concurrent.

Marin Ionescu