# Construction of Steiner Porism with concentric circles

I tried to construct Steiner chain in geogebra with 2 concentric circles. I used the radius 2 for big circle and 0.5 for iside circle. So, radius of steiner circles is 0.75. After construction i get as result that 2 circles of my chain aren't tangent. I don't really understand a link between circles radius'. I would like to understand why it doesn't work. Here is an image. Wikipedia gives a feasibility criterion for a Steiner chain of $$n$$ circles to be supported between two concentric circles of radii $$R$$ and $$r$$: $$\frac{1+\sin\pi/n}{1-\sin\pi/n}=\frac Rr$$ The parameters you have chosen do not satisfy this relation.
By solving the above equation for $$n$$, we can determine for given $$R$$ and $$r$$ whether there is any Steiner chain between the circles: $$r(1+\sin\pi/n)=R(1-\sin\pi/n)$$ $$(R+r)\sin\pi/n=R-r$$ $$\sin\frac\pi n=\frac{R-r}{R+r}$$ The criterion here is that $$n$$ must be an integer at least $$3$$, or at least a rational number $$\frac pq$$ corresponding to a chain with $$p$$ circles that loops $$q$$ times before closing. In particular, if $$R$$ and $$r$$ are rational, then by Niven's theorem $$n=6$$ and $$\frac{R-r}{R+r}=\frac12$$, implying $$R=3r$$. But in the diagram in the question $$R=2$$ and $$r=\frac12$$, which does not satisfy $$R=3r$$, so no Steiner chain of any number of circles can exist between these two circles.
If we want to have a Steiner chain of $$n=5$$ circles with the outer concentric circle having radius $$R=2$$, the first feasibility criterion tells us that the inner concentric circle's radius $$r$$ must be $$\frac{2(1-\sin\pi/5)}{1+\sin\pi/5}=22-8\sqrt5-4\sqrt{50-22\sqrt5}=0.519232367\dots$$ 