[b]Proposition:[/b] To inscribe a chain of tangent circles in the figure known as the Arbelos (shoemaker's knife).
Solution; vector reduction.
Giving the kth circle in terms of the ellipse parameters, and k.
_____________________
Archimedes' Arbelos:
[list]
[*]1a. Inscribe a circle in the arc.[url]http://www.geogebratube.org/material/show/id/54105[/url]
[*]1b. Tangent circles in the arc (Solution 1).
[*][b]→1c. Vector reduction[/b]
[*]1d. Ellipse from one parameter, scale and rotation: [url]http://www.geogebratube.org/material/show/id/55256[/url]
[*]1e. Final Construction: [url]http://www.geogebratube.org/material/show/id/54592[/url]
[*]2a. Let one circle enclose another.
Inscribe a third circle in the ring: [url]http://www.geogebratube.org/material/show/id/54595[/url]
[*]2b. Tangent circles in the ring. [url]http://www.geogebratube.org/material/show/id/54596[/url]
[/list]
3. Cyclic Solution:
[list]
[*]3a. An outer ring of tangent circles: [url]http://www.geogebratube.org/material/show/id/55009[/url]
[*]3b. Determine the projection.
[*]3c. Final Construction: [url]http://www.geogebratube.org/material/show/id/55883[/url]
[/list]
