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1. INTRODUCTION. This paper has its genesis in a problem the author first came upon while in college. Although the areas covered here are well travelled and nothing here is guaranteed original, it covers a pleasant nexus of many mathematical strands. Furthermore, we show the value of good notation and of reading an old master for solving a problem. Consider… (More)

We present a short proof of Descartes Circle Theorem on the " curvature-centers " of four mutually tangent circles. Key to the proof is associating an octahedral configuration of spheres to four mutually tangent circles. We also prove an analogue for spheres. It can be traced back to at least Descartes that four mutually tangent circles have curvatures… (More)

We present a short new proof that the continued fraction of a quadratic irrational eventually repeats. The proof easily generalizes; we construct a large class of functions which, when iterated, must eventually repeat when starting with a quadratic irrational.

Let p be a quadratic polynomial over a splitting field K, and S be the set of zeros of p. We define an associative and commutative binary relation on G ≡ K ∪ {∞} − S so that every Möbius transformation with fixed point set S is of the form x " plus " c for some c. This permits an easy proof of Aitken acceleration as well as generalizations of known results… (More)

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