# Thread: Nested Tangent Circles

1. ## Nested Tangent Circles

I need help with the following problem.

All circles are tangent to each other, if the radius of the small circle is r then what is the radius of the largest one in terms of r? I can visually see that it is 6r, but I want to know why.

I've tried constructing several triangles in the picture, but nothing clicks.

2. ## Re: Nested Tangent Circles

If we let $x$ be the distance from the center of the largest circle (radius $R$) to the center of the smallest circle (radius $r$), and observing that the medium sized circle has radius $\frac{R}{2}$, we may use the Pythagorean theorem to state:

$\left(\frac{R}{2} \right)^2+x^2=\left(\frac{R}{2}+r \right)^2$

and we see that:

$R=x+r=\sqrt{r(r+R)}+r$

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

$R^2-2rR+r^2=r^2+rR$

$R^2=3rR$

For $R\ne0$, we find:

$R=3r$