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, Myshkin May 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.

  • 1
    $\begingroup$ It is not clear what you denote by $r_{ext}$ and $r_{int}$. Please define/describe the problem precisely, including notation. $\endgroup$ – 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.

  • $\begingroup$ I've already done that. I could show you the complete formula for radii instead curvatures. My calculations seems right I don't know where my mistake is. $\endgroup$ – Davide Melfi May 16 '16 at 16:08
  • $\begingroup$ Mathematica disagrees... $\endgroup$ – Igor Rivin May 16 '16 at 16:33
  • $\begingroup$ I imagine that, but I uninstalled mathematica a long time ago, i'm more interested in what my mistake was. $\endgroup$ – Davide Melfi May 16 '16 at 16:39

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.


Not the answer you're looking for? Browse other questions tagged or ask your own question.