Let A and B be two fixed points on a straight line. Two circles touch this line at A and B respectively and the tangent to each other at M, when the circles vary the locus of M is?
This question has already been asked here, but I could not comprehend that answer properly, also I am not allowed to comment on that answer as I don't have 50+ reputation. I have only drawn the rough sketch but not able to go any further.
The locus of $M$ is the circle $\Gamma$ diameter $AB$.
Any circle $\Gamma_B$ touching the line $AB$ at $B$ is orthogonal to $\Gamma$ at $B$ and hence also at its other point of intersection (by reflection in the line $OO_B$). Call this point $M$. $OM$ is tangent to $\Gamma_B$ at $M$.
Let the bisector of $\angle AOM$ meet at $O_A$ the perpendicular to $AB$ at $A$. Then $O_AM$ is equidistant from $OA$ and $OM$, so the circle centre $O_A$ touching $AB$ at $A$ also touches the circle $\Gamma_B$ at $M$.