Four Mutually Tangent & Exterior Circles
- Ryan Hirst
Proposition: To determine four mutually tangent, mutually exterior circles.
You, unwary observer, may have stumbled into the middle of an argument. This is a special case of Apollonius' Tangency Problem. For more information about the method used in this worksheet, a complete solution is here: http://www.geogebratube.org/material/show/id/34645.
- Bounded Arcs: The limiting position of circle C occurs when its boundary passes into a tangent line to circles A, B. A common tangent to A,B, intersects midline AB at a Similarity Point (http://www.geogebratube.org/material/show/id/34182).
- Point C:Given two tangent circles, the locus of the third center can be stated as: Find the locus of points equidistant from two circles. With the condition that the circles be mutually exterior, and tangent. The resulting locus is one branch of a hyperbola: http://www.geogebratube.org/material/show/id/27216 (the orange solution).
- Similarity Axis: A Similarity axis is a straight line through any two similarity points of the circles A,B,C. Only one axis satisfies the constraints of this worksheet.
- is the Power Center of the first three circles. A construction (Monge's problem) is here: http://www.geogebratube.org/material/show/id/33929.
- Points of Tangency: Following the method outlined in the Apollonius worksheet, the points of tangency of the final (fourth) circle D can be constructed as follows: For each circle, take the closest point on the Similarity Axis, and then draw the conjugate point (like this: http://www.geogebratube.org/material/show/id/34578). Draw a line joining the conjugate point to the Power Center. Where this line intersects the given circle, the new circle will be tangent.
- 1. Tangent along the rim: solve for k
- 2a. Initial position: http://www.geogebratube.org/material/show/id/58360
- 2b. Tangent to equal circles: http://www.geogebratube.org/material/show/id/58455
- →3a. Four mutually tangent & exterior circles (Apollonius)
- 3b. Vector reduction: http://www.geogebratube.org/material/show/id/58461
- Affine Transformation http://www.geogebratube.org/material/show/id/58177
- Reflection: Line about a Circle http://www.geogebratube.org/material/show/id/58522
- Reflection: Circle about a Circle http://www.geogebratube.org/material/show/id/58185
- Circle Inversion: The Metric Space http://www.geogebratube.org/material/show/id/60132
- Sequences 1: Formation http://www.geogebratube.org/material/show/id/58896
- Sequence 1: Formation http://www.geogebratube.org/material/show/id/59816
- Sequence 1: Iteration 1 http://www.geogebratube.org/material/show/id/59828
- Example of equivalent projections: http://www.geogebratube.org/material/show/id/65754
- Final Diagram: http://www.geogebratube.org/material/show/id/65755