The problem is from the image: the yellow and the red and tangents, so are the blue and the red and the yellow passes through that black thick dot at the north pole of the blue one.
So my attempt:
I had no idea on what to do, maybe some homotheties could help, but not directly… so I did the analytic calculation. Put the $x$ axis on the common tangent line and the origin at the tangency point between red and blue ones.
Call $a$ the radius of the red circle, $b$ the radius of the blue one. $b>a$
So, if $A =(t,0)$, with $t<0$, is the center of the yellow one, then I found:
$t = -\frac{b}{2a} \sqrt{b^2-4a^2}$
which is quite weird because it needs $b>2a$ which clearly is not the case in the figure.
I can't see a mistake in the calculations, the line joining $A$ and the center of the red circle $(0,a)$ is:
$\frac ya + \frac xt = 1$
it is not that hard to find the contact point of the yellow and red one…
tldr: my calculations look weird and I got no synthetic idea of how to solve this one.
Best Answer
Let $NS$ be the diameter of the blue circle through the common tangency point $S$. Draw a circle centered at $N$ whose radius is the red circle’s diameter; let it meet the blue circle at $X$. Let $NX$ meet the common tangent at $A$. Then the desired semicircle has center $A$ and radius $AN$.
Proof: $AO^2 = AS^2 + OS^2 = AN⋅AX + OS^2 = AN(AN - 2OS) + OS^2 = (AN - OS)^2$.