The mixtilinear touch point, the incenter and an arc midpoint
Problem
Let $ABC$ be an acute triangle with incenter $I$, and let $T$ be the point where the $A$-mixtilinear incircle touches the circumcircle. Let $M_b$ and $M_c$ be the midpoints of the minor arcs $AC$ and $AB$, and let $N_a$ be the midpoint of the major arc $BC$. Prove that $N_a$, $I$, $T$ are collinear and that the quadrilateral $AM_cTM_b$ is harmonic.
The diagram
Drag any point and the whole construction follows. Tap a point to pin it, double-click to go back to the start. Effects redraws the figure in grayscale for print, in colour, or with the givens in focus.