Descartes' theorem and Circle Packing [closed]

Question

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 r_ext is the external radius and r_int 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?

Answer 1

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.

Answer 2

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.