Ptolemy's theorem, found by Claudius Ptolemy (ca. 100-170 CE), was a key mathematical achievement in antiquity that allowed for the calculation of numerical values of trigonometric functions. It states that for any quadrilateral inscribed into a circle, the product of its diagonals, p·q, is equal to the sum of the two products of opposite sides, a·c + b·d. This can be easily seen by recognizing that the two red triangles in the left panel are similar to each other, as are the two blue triangles in the right panel (watch the animation in the middle panel and remember the Inscribed Angle Theorem). Thus, we find the two relations shown in the yellow insets. By adding both equations and multiplying the result with p, we find the theorem.