Conic section by trisecting an arc of a variable circle through two points

Question

Given two distinct point $A$ and $B$, consider arc $\overset{\huge\frown}{AB}$ of a variable circle through those points. Points $M$ and $M^\prime$ trisect this arc; specifically, $|\overset{\huge\frown}{AM}|= |\overset{\huge\frown}{MM^\prime}| = |\overset{\huge\frown}{M^\prime B}|$. Let $d$ be the perpendicular bisector of $\overline{AB}$, and let $I$ be the midpoint of $\overline{MM^\prime}$.

Prove that $M$ belongs to a fixed conic of directrix $d$.

Answer 1

You can easily prove that $MA=2MI$, that is the distance from $M$ to point $A$ is twice the distance from $M$ to line $D$. It follows that $M$ belongs to a hyperbola of directrix $D$ and focus $A$.