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Grade Vs Difficulty:

Teachers will begin with the famous problem of Apollonius, to find circles that are tangent to three given circles, and to count how many such circles there are. Then we talk about how Descartes answered a question asked of him by Princess Elisabeth of Bohemia in which he finds the radius of a circle that is tangent to three given mutually tangent circles, using his (new) techniques of algebraic geometry. Finally, we talk about the Apollonian Gasket, a fractal that is generated by mutually tangent circles, the radii of which can be found using Descartes' formulas.
Extensions include talking about rational numbers (in terms of solutions that arise), and possibly even Ford circles, in the case where the outer "circle" is really a straight line and two unit circles are tangent to this line at the numbers 0 and 1, say.