KöMaL 2019 B5025

The inscribed circle of triangle $ABC$ is centred at $I$, and touches sides $BC$, $CA$ and $AB$ at points $D$, $E$ and $F$, respectively. Let $M$ be an arbitrary point in the interior of side $BC$, different from $D$. Let the lines $DI$ and $EF$ intersect at $T$, and let $K$ denote the midpoint of line segment $MT$. Prove that the circles $DEF$, $TDM$ and $KIT$ are concurrent.