Explore Ptolemy's theorem and its generalizations in this mathematics lecture from the Rational Geometry Math Foundations series. Delve into the classical theorem about cyclic quadrilaterals, reframing it from a rational point of view to create a purely algebraic formulation without relying on concepts of distance. Discover how the Triple Quad Formula aids in this process and learn about the theorem's broader applications in finite fields and relativistic planar geometry. Gain insights into how reformulating classical Euclidean geometry results without infinite processes can lead to more general and wide-ranging mathematical discoveries.