Four Mutually Tangent & Exterior Circles

Author:
Ryan Hirst
Proposition: To determine four mutually tangent, mutually exterior circles.
GeoGebra Applet
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.
  1. 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).
  2. 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).
  3. 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.
  4. is the Power Center of the first three circles. A construction (Monge's problem) is here: http://www.geogebratube.org/material/show/id/33929.
  5. 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.
Let me know if more information would be of assistance. I am always happy to clarify my assumptions, and demonstrate them with worksheets. ______________ The Tangent Circle Problem: Solution:

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