KöMaL 2022 A817

Let $ABC$ be a triangle. Let $T$ be the point of tangency of the circumcircle of triangle $ABC$ and the $A$-mixtilinear incircle (the circle which is tangent to sides $AB$, $AC$, and internally tangent to the circumcircle of triangle $ABC$). The incircle of triangle $ABC$ has center $I$ and touches sides $BC$, $CA$ and $AB$ at points $D$, $E$ and $F$, respectively. Let $N$ be the midpoint of line segment $DF$. Prove that the circumcircle of triangle $BTN$, line $TI$ and the perpendicular from $D$ to $EF$ are concurrent.