I had $3$ circles of radii $1$, $2$, $3$, all touching each other. A smaller circle was constructed such that it touched all the $3$ circles.

What is the radius of the smaller circle?

This is what I did:

I conveniently positioned the $3$ circles on the coordinate axes and found the coordinates of the centers of each circle.

Then I wrote a general equation of a circle( for the smaller one) , and using the fact that distance between the centers is equal of the sum of radii ( for circles touching each other) I found $3$ equations, which I could use to solve for the variables in the general equation. Hence I found the equation of the smaller circle, and thus its radius.

However, I believe that this method is very inefficient as I ended up with so many steps and sub steps to solve the equations.

Is there a better way to approach this?

I would prefer a geometrical solution instead of a coordinate solution.

Thanks for the help!!

Note :

I found that this question already has an answer here:https://math.stackexchange.com/questions/1867315/problem-including-three-circles-which-touch-each-other-externally

However is there a more efficient solution for this than what was mentioned in those answers?

  • $\begingroup$ I would advise you to have a look at the complex version of Descarte"s theorem here $\endgroup$ – Jean Marie Jun 12 at 8:42
  • $\begingroup$ But if you want coordinate free proofs,, look at the more general "Apollonius problem" $\endgroup$ – Jean Marie Jun 12 at 8:47

Let $A$, $B$ and $C$ be centers of circles with radius $3$, $2$ and $1$ respectively and let $x$ be a radius of the needed circle with a center $D$.

Thus, $$\measuredangle ACB=90^{\circ},$$ $$\cos\measuredangle ACD=\frac{4^2+(1+x)^2-(3+x)^2}{2\cdot4(1+x)}=\frac{2-x}{2(1+x)},$$

$$\cos\measuredangle BCD=\frac{3^2+(1+x)^2-(2+x)^2}{2\cdot3(1+x)}=\frac{3-x}{3(1+x)},$$ which gives $$\left(\frac{2-x}{2(1+x)}\right)^2+\left(\frac{3-x}{3(1+x)}\right)^2=1$$ or $$23x^2+132x-36=0,$$ which gives $$x=\frac{6}{23}.$$

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    $\begingroup$ Thank you so much! $\endgroup$ – Vamsi Krishna Jun 12 at 8:24

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Notice, the radius $r$ of the small (inscribed) circle externally touching any three externally kissing (touching) circles of radii $a, b$ & $c$ is given by the generalized formula as follows

$$\boxed{\color{blue}{r=\frac{abc}{2\sqrt{abc(a+b+c)}+ab+bc+ca}}}$$ Now, substituting the values of radii of three externally touching circle i.e. $a=1, b=2$ & $c=3$ in above generalized formula, we get radius of small circle

$$r=\frac{1\cdot 2\cdot 3}{2\sqrt{1\cdot2\cdot 3(1+2+3)}+1\cdot 2+2\cdot 3+3\cdot 1}$$ $$r=\color{blue}{\frac{6}{23}}$$

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    $\begingroup$ Thanks a lot!!! $\endgroup$ – Vamsi Krishna Jun 12 at 8:24

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