Ratio of outer circle to inner circle diameter in pentagon star I am trying to write a code to draw pentagon star and have stumbled upon a problem. In the attached diagram, what is the ratio of the diameter of outer (red) circle to the inner (blue) circle. The circles are concentric. I've read that it is ~2.61 but would like to develop the relationship myself.

 A: Hint: everything can be done with analytic geometry.
If the outer radius is $R$, then in a cartesian coordinate system the vertices on the outer circle are at $[0,R], \ldots$.  Find the
equations of the lines, see where they intersect...
A: @Robert Israel :  Here is an alternate proof, involving only triangle trigonometry.  
With notations displayed on the figure below, we have to find the ratio $\dfrac{OA}{OB}.$


*

*by reasoning in right triangle $OCA \ $: $ \ \widehat{OAB}=90°-72°=18°.$

*by reasoning in right triangle $OCB \ $: $ \ \widehat{OBC}=90°-72°/2=54°$, thus $\widehat{ABO}=180°-54°=126°.$
Thus, sine law in triangle $OAB$ gives:
$$\tag{1}\dfrac{OA}{OB}=\dfrac{\sin \widehat{ABO}}{\sin \widehat{OAB}}=\dfrac{\sin 126°}{\sin 18°}=\dfrac{\cos 36°}{\cos 72°}$$
 (exact value); thus $\dfrac{OA}{OB}\approx 2.618.$
Knowing that $\cos 36°=\dfrac{1+\sqrt{5}}{4}$, it is not difficult to conclude that the exact value can also be written$1+\Phi$ (where $\Phi:=\dfrac{1+\sqrt{5}}{2}$ is the golden ratio).

