[b]Proposition:[/b] the image of a circle... is a circle.
For the proof I used algebra and observation. Procedure: By definition, circle reflection in the away direction is projection on a line. (Divide one pie into 12 pieces. One piece is 1/12 of a pie. There is nothing more to inversion) Project the endpoints of the diameter lying on line AB. The projection of the midpoint of circle lies on this same line. Call the projected figure, whatever it is, a Garmupp Call the "diameter" of the Garummpp any unbroken, finite line (curved or straight) which is the projection of a chord of circle β on a secant line from A. I know that the diameter of the Garumpp is a straight segment when the secant line is AB. Suppose I think the Garumpp is a circle. If so, its center is the midpoint of this diameter. The question can be asked directly in algebra. The hunch turns out to be true. It should be clear that the whole figure is projected. Show me an unbroken part of the arc β, and I will consider a formal proof that the projection is a whole circle. But I have already decided there is no such point (the circle is continous). QED. _________________ 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] [*]2b. Tangent to equal circles: [url]http://www.geogebratube.org/material/show/id/58455[/url] [*]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] [*][b]→Reflection: Circle about a Circle[/b] [*]Circle Inversion: The Metric Space [url]http://www.geogebratube.org/material/show/id/60132[/url] [/list] Iteration: [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]