Proposition:
[i]Given two tangent circles of equal radii, to inscribe a chain of circles tangent to both.[/i]
The locus of centers is a straight line.
If the radii are unequal, the locus is one branch of a hyperbola. I don't want to solve that problem. Instead, I will solve this (simpler) problem, and project the solution (the chain of circles) back into my diagram.
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The Tangent Circle Problem:
[list]
[*]1. Tangent along the rim: solve for k
[*]2a. Initial position: [url]http://www.geogebratube.org/material/show/id/58360[/url]
[*][b]→ 2b. Tangent chain to equal circles[/b]
[*]3a. Four mutually tangent & exterior circles (Apollonius): [url]http://www.geogebratube.org/material/show/id/58189 [/url]
[*]3b. Vector reduction: [url]http://www.geogebratube.org/material/show/id/58461[/url]
[/list]
[list]
[*]Affine Transformation [url]http://www.geogebratube.org/material/show/id/58177[/url]
[*]Reflection: Line about a Circle [url]http://www.geogebratube.org/material/show/id/58522[/url]
[*]Reflection: Circle about a Circle: [url]http://www.geogebratube.org/material/show/id/58185[/url]
[*]Circle Inversion: Metric Space: [url]http://www.geogebratube.org/material/show/id/60132[/url]
[/list]
Solution:
[list]
[*]Sequences 1: Formation [url]http://www.geogebratube.org/material/show/id/58896[/url]
[*]Sequence 1: Formation [url]http://www.geogebratube.org/material/show/id/59816[/url]
[*]Sequence 1: Iteration 1 [url]http://www.geogebratube.org/material/show/id/59828[/url]
[*]Example of equivalent projections: [url]http://www.geogebratube.org/material/show/id/65754[/url]
[*]Final Diagram: [url]http://www.geogebratube.org/material/show/id/65755[/url]
[/list]
