Construct circle internally tangent to a larger circle, and tangent to a point on a chord of the larger circle

Question

Given a larger circle $O$ and its chord $AB$, construct circle $P$ that is internally tangent to $O$ and and tangent to point $C$ on $AB$.

The chord and the point on it are completely arbitrary.

This could be a variant of the Apollonius's problem, but I've no time to sift through the literature. Or this could just be something entirely simple. I have no thoughts to show as I am completely stumped by the problem.

Answer 1

You could try first solving it algebraically, taking the radius of the circle and the distance $BC$ as knowns and the radius of circle $P$ as the unknown to be solved. This gives you a formula for the radius of $P$ in terms of the other lengths, which you can then use to construct the radius from those segments. There might be a special insight that will solve this more directly, but this will get the job done.