Construction of Steiner Porism with concentric circles

Question

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.

Answer 1

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.

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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 $$\frac{2(1-\sin\pi/5)}{1+\sin\pi/5}=22-8\sqrt5-4\sqrt{50-22\sqrt5}=0.519232367\dots$$