# Descartes' theorem and Circle Packing [closed]

There's something I am missing comparing Descartes' theorem for three isometric circles here and this wiki post on circle packing of 3 circles here.

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?

## closed as off-topic by Stefan Kohl, Yoav Kallus, Chris Godsil, Jan-Christoph Schlage-Puchta, MyshkinMay 16 '16 at 22:37

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question does not appear to be about research level mathematics within the scope defined in the help center." – Chris Godsil, Jan-Christoph Schlage-Puchta
If this question can be reworded to fit the rules in the help center, please edit the question.

• It is not clear what you denote by $r_{ext}$ and $r_{int}$. Please define/describe the problem precisely, including notation. – GH from MO May 16 '16 at 15:51

The $k$s in the Wiki article are the curvatures (the inverses of the radii). If you correct for that, the two numbers will agree.
$$1+2\frac{\sqrt{(3)}}{3}$$