Rational Steiner porism (Q2874421)

Let \(I(r)\) be a circle in the interior of another circle \(O(R)\). Denote by \(d\) the distance between their centres and put \(q=\frac{(R-r)^2-d^2}{4rR}\). It is well known that there exists a closed chain of \(n\) mutually tangent circles each tangent internally to \((O)\) and externally to \((I)\) if and only if \(q=\tan ^2 \frac{\pi}{n}\). Such a configuration is known as Steiner \(n\)-cycle. The author first gives formulae relating the radii of neighbouring circles in a Steiner chain and points out an easy construction of the neighbor knowing the tangency point. Next the author studies rational Steiner configurations, viz., those in which \(R\), \(r\), \(d\) are given by rational numbers. It is easy to see that rational Steiner \(n\)-chains exist only for \(n=3\), \(4\), \(6\). Specialization of previous results leads to parametrization for all triples \((R,r,d)\) with \(R=1\) corresponding to a rational Steiner chain. The paper concludes with a brief examination of relations among rational Steiner chains with different parameters.