For more complicated problems I find it easier to give figures in position, bound in the unit circle, solve the desired relationships, and then scale and rotate the whole thing.

_____________________[br]Archimedes' Arbelos:[br][list][br][*]1a. Inscribe a circle in the arc.[url]http://www.geogebratube.org/material/show/id/54105[/url] [br][*]1b. Tangent circles in the arc (Solution 1).[br][*]1c. Vector Reduction: [url]http://www.geogebratube.org/material/show/id/54557[/url][br][*][b]→1d. Proposition: To give an ellipse by one parameter, scale and rotation.[/b][br][*]1e. Final Construction: [url]http://www.geogebratube.org/material/show/id/54592[/url][br][br][*]2a. Let one circle enclose another.[br] Inscribe a third circle in the ring: [url]http://www.geogebratube.org/material/show/id/54595[/url] [br][*]2b. Tangent circles in the ring. [url]http://www.geogebratube.org/material/show/id/54596[/url] [br][/list][br]3. Cyclic Solution:[br][list][br][*]3a. An outer ring of tangent circles: [url]http://www.geogebratube.org/material/show/id/55009[/url] [br][*]3b. Determine the projection.[br][*]3c. Final Construction: [url]http://www.geogebratube.org/material/show/id/55883[/url][br][/list]