As an old school draughtsman, I’m surprised that nobody has suggested how to go about constructing two circles which intersect at a known tangent point. I’m also a little puzzled that nobody has asked why you are trying to construct the geometry that you require, as the “why” may suggest a particular “how”.
Going back to your initial diagram, you have a circle, or arc (A), and assuming that you know where you want to create the second circle (B) such that they intersect at a known point, then simply do the following.
Create a guide from the centre of circle (A) to the desired intersection point (tangent point).
Next, offset circle (A) by the radius of the second circle (B) to create a new circle ©. Where circle © intersects with the guide is the centre point of circle (B).
If you want to get picky about it, you can re-draw the circles such that the “points” on the polygon (circle) meet at the intersection / tangent point.
One of these days, I’ll get round to learning how to create the nice little animations, but for now you’ll have to make do with an image …