3 circles internal tangent 
My friend show me the diagram above , and ask me 
"What is the area of a BLACK circle with radius of 1 of BLUE circle?"
So, I solved it by algebraic method.
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Let center of  $\color{black}{BLACK}$ circle be $(0,0)$.
We can set, 
$x^2 + (y-R)^2 = R^2$ , where $R$ means radius of $\color{red}{RED}$ circle.
$(x-p)^2 + (y-r)^2 = r^2 $,  where $(p,r)$ means center of $\color{blue}{BLUE}$ circle.
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These can imply
$    2R=r+ \sqrt{p^2 + r^2}$
$p^2 + (R-r)^2 = (R+r)^2 $
So, 
$ 2r=R$
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But he wants not algebraic but Geometrical Method.
How can I show  $ 2r=R$ with Geometrical Method?
Really thank you.
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(Actually I constructed the diagram with algebraic methed, 
but I'd like to know how construct this whit Geometrical method.)

I you perform a circular inversion w.r.t the black circle, the red circle becomes the red tangent line in the picture below, while the blue circle gets reflected to another circle, tangent to the black circle, the black line and the red line. The diameter of this new circle must be $1$. Therefore $\overline{AB}=1+1=2$ and $\overline{AC}=1/\overline{AB}=1/2$.

