An Alternative Proof of Ptolemy's Theorem via Differential Equations

We present an alternative proof of Ptolemy's Theorem using an ordinary differential equations (ODE) framework. By parameterizing an inscribed cyclic tetragon and utilizing the chord length formula in polar coordinates, we successfully bypass the traditional need for intricate trigonometric identities. The geometric problem is thus reduced to analyzing a second-order linear homogeneous ODE subjected to specific physical boundary conditions. Through the uniqueness of the trivial solution, we globally establish the classical relation for the products of the sides and diagonals, highlighting an exotic and highly pedagogical connection between elementary geometry and differential calculus.