Does the sequence of circumcircle mid-arc triangles converge to an equilateral triangle?

Question

This is a follow-up question on The triangle of intersections of angle bisectors and perpendicular bisectors of a triangle

Let $\triangle_1$ be a triangle,
$\triangle_2$ the circumcircle mid-arc triangle of $\triangle_1$,
$\triangle_3$ the circumcircle mid-arc triangle of $\triangle_2$,
...
$\triangle_n$ the circumcircle mid-arc triangle of $\triangle_{n-1}$.

The sequence seems to converges to an equilateral triangle.
Is this true?
How can we prove it?