There's something I am missing comparing Descartes' theorem for three isometric circles [here](https://en.wikipedia.org/wiki/Descartes%27_theorem) and this wiki post on circle packing of 3 circles [here](http://mathworld.wolfram.com/CirclePacking.html).


From my calculation:


$$
r\_{ext} = \frac{r\_{int}}{{(3\pm2\sqrt3)}}, \tag{1}
$$


where *rext* is the external radius and *rint* the internal radius. In the second article it seems:


$$
r\_{ext} = r\_{int}(1+2\frac{\sqrt3}{3}). \tag{2}
$$


(Probably is crappy mathematics of my own.)


For the sake of completness. The generic formula I derived from Descartes' theorem is (for different radii):


$$
r\_{ext} = \frac{r\_{1}r\_{2}r\_{3}}{(r\_{1}r\_{2}+r\_{2}r\_{3}+r\_{1}r\_{3} \pm 2{\sqrt{r\_{1}r\_{2}r\_{3}(r\_{1}+r\_{2}+r\_{3})}{}})}. \tag{3}
$$


what am I missing?



Descartes' theorem and Circle Packing
My bad. I misunderstood the second article:


$$
1+2\frac{\sqrt{(3)}}{3}
$$


is the coefficient in a linear expression. Both the solutions are the same, you can derive (2) from (1) using square difference formula.


