enter image description here

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. $$$$

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. $$$$ These can imply

$ 2R=r+ \sqrt{p^2 + r^2}$

$p^2 + (R-r)^2 = (R+r)^2 $


$ 2r=R$


But he wants not algebraic but Geometrical Method.

How can I show $ 2r=R$ with Geometrical Method?

Really thank you.


(Actually I constructed the diagram with algebraic methed,

but I'd like to know how construct this whit Geometrical method.)


enter image description here

\begin{align} \triangle ADB:\quad |DB|^2&= (\tfrac{R}2+r)^2 -(\tfrac{R}2-r)^2 =2rR ,\\ \triangle BDO:\quad |DB|^2&= (R-r)^2-r^2 =R(R-2r) . \end{align}

\begin{align} R(R-2r)&=2rR ,\\ R&=4r . \end{align}

  • $\begingroup$ So nice! Really thank you. $\endgroup$ – user143993 Feb 5 at 6:46

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 $\overline{AC}=1/\overline{AB}=1/2$.

enter image description here

  • $\begingroup$ Really thank you. $\endgroup$ – user143993 Oct 27 '18 at 11:18

Based on the figure, one can make some inspired guesswork.

Construct a rectangle $ABCD$ with side $AB$ of length $1$ and diagonal $AC$ of length $3.$ Extend $AB$ to $E$ so that $B$ between $A$ and $E$ and $AE = 2.$

Construct a circle of radius $4$ about $A,$ a circle of radius $2$ about $E,$ and a circle of radius $1$ about $C.$

Confirm that the circles about $A$ and $E$ are internally tangent, the circles about $A$ and $C$ are internally tangent, and the circles about $C$ and $E$ are externally tangent. Extend the side $AD$ to a diameter of the circle about $A$ and confirm that this diameter is tangent to both the circle about $C$ and the circle about $E.$

Therefore the figure composed of these three circles and this diameter is congruent to the given figure, and the circles about $A,$ $C,$ and $E$ correspond respectively to the black, blue, and red circles.

  • $\begingroup$ I appreciate you. Really thank you. $\endgroup$ – user143993 Oct 27 '18 at 11:18

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