Abstract 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 to write down the roots of the Lucas polynomials.

中文翻译：

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椭圆的弦、卢卡斯多项式和三次方程

摘要 Thomas Price 的一个美丽定理将斐波那契数和卢卡斯多项式与椭圆的平面几何联系起来，概括了一个关于圆的经典问题。我们简要介绍了圆问题的历史，说明了普莱斯的椭圆证明，以及重新组织的证明，其中包含一些旨在将结果置于与经典数学的紧密联系网络中的新想法。它的灵感来自卡尔达诺的三次方程解和牛顿关于幂和的定理，并根据对称多项式理论对广义卢卡斯多项式进行了解释。我们还开发了沿途浮出水面的其他联系；例如，我们给出了广义斐波那契多项式的平行解释，并证明了卡尔达诺的方法可用于写出卢卡斯多项式的根。