Ptolemy's theorem

Depends on: AA similarity (AA similarity), Inscribed angles on same arc are equal; angle in a semicircle is right (inscribed angles on the same arc are equal).

Statement

Let quadrilateral $ABCD$ be inscribed in a circle (i.e. the four vertices lie on a common circle), arranged in order; let the two diagonals be $AC$ and $BD$. Then the product of the two diagonals equals the sum of the products of the two pairs of opposite sides:

This is the celebrated Ptolemy's theorem. It translates the topological fact "four points concyclic" into an algebraic identity among six segment lengths.