Ptolemy's theorem

Statement

Let $A$, $B$, $C$ and $D$ be concyclic points of the Euclidean plane $\mathbb{R}^2$, and let $d$ be the Euclidean distance. Suppose that the segments $AC$ and $BD$ meet in a point that is inside both segments. Thus $ABCD$ is a convex cyclic quadrilateral with the diagonals $AC$ and $BD$. Then $$d(A, C)\, d(B, D) = d(A, B)\, d(C, D) + d(B, C)\, d(A, D).$$

Remarks

In words: the product of the diagonals is the sum of the products of the opposite sides. The converse is also true. If a quadrilateral has this property, then it is cyclic. For each four points of the plane, $d(A, C)\, d(B, D) \leq d(A, B)\, d(C, D) + d(B, C)\, d(A, D)$. This is Ptolemy's inequality. For a rectangle with the sides $a$ and $b$, the theorem gives $a^2 + b^2 = c^2$, with the diagonal $c$. This is the Pythagorean theorem.

History

The theorem has the name of the astronomer Ptolemy. He used it to make his table of chords, a trigonometric table for astronomy.

Explanations

L1 human

Put four points in order on a circle and join them into a quadrilateral. Multiply the two diagonals. You get the same number as when you multiply the opposite sides in pairs and add the two products.