Help on how to solve this question involving tangent circles and semicircles

Question

Hi,

I gave a link to the problem. I’ve tried multiple times to solve it but can’t. Can someone provide help please?

Edit: In case it wasn’t clear , I’m referring to the top question

Answer 1

If $x$ is the radius of the circle with center $P$, then: $$AP=2+x,\quad BP=1+x,\quad CP=3-x.$$ If $H$ is the projection of $P$ on $AB$, we can also set: $$PH=h,\quad AH=a,\quad BH=3-a,\quad CH=a-1.$$ From Pythagoras' theorem applied to triangles $APH$, $BPH$, $CPH$ you can then obtain three equations in the three unknowns $x$, $h$, $a$, which can be easily solved.