The Problem of Apollonius as an Opportunity for Teaching Students to Use Reflections and Rotations to Solve Geometry Problems via Geometric (Clifford) Algebra

Authors: James A. Smith

Note: The Appendix to this new version gives an alternate--and much simpler--solution that does not use reflections. The beautiful Problem of Apollonius from classical geometry ("Construct all of the circles that are tangent, simultaneously, to three given coplanar circles") does not appear to have been solved previously by vector methods. It is solved here via Geometric Algebra (GA, also known as Clifford Algebra) to show students how they can make use of GA's capabilities for expressing and manipulating rotations and reflections. As Viète did when deriving his ruler-and-compass solution, we first transform the problem by shrinking one of the given circles to a point. In the course of solving the transformed problem, guidance is provided to help students ``see" geometric content in GA terms. Examples of the guidance that is given include (1) recognizing and formulating useful reflections and rotations that are present in diagrams; (2) using postulates on the equality of multivectors to obtain solvable equations; and (3) recognizing complex algebraic expressions that reduce to simple rotations of multivectors. As an aid to students, the author has prepared a dynamic-geometry construction to accompany this article.

Comments: 18 Pages.

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Submission history

[v1] 2016-05-22 20:06:09
[v2] 2016-06-03 19:48:48
[v3] 2016-06-05 15:58:39
[v4] 2016-06-11 07:43:18
[v5] 2016-08-20 21:29:55

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