  A beautiful theorem of Thomas Price links the Fibonacci numbers and the Lucas
polynomials to the plane geometry of an ellipse, generalizing a classic problem
about circles. We give a brief history of the circle problem, an account of
Price's ellipse proof, and a reorganized proof, with some new ideas, designed
to situate the result within a dense web of connections to classical
mathematics. It is inspired by Cardano's solution of the cubic equation and
Newton's theorem on power sums, and yields an interpretation of generalized
Lucas polynomials in terms of the theory of symmetric polynomials. We also
develop additional connections that surface along the way; e.g., we give a
parallel interpretation of generalized Fibonacci polynomials, and we show that
Cardano's method can be used write down the roots of the Lucas polynomials.
