GeoProof — Ptolemy's Theorem explorer

AC · BD = AB · CD + AD · BC

Drag any vertex around the circle — the four points stay in order, so the quadrilateral is always cyclic and the equation holds exactly.

Ptolemy's Theorem

Take any four points A, B, C, D lying on a circle, in order, forming a cyclic quadrilateral. Ptolemy's theorem relates its two diagonals to its four sides:

AC · BD = AB · CD + AD · BC

The product of the diagonals equals the sum of the products of the two pairs of opposite sides.

A quick check: the rectangle

For a rectangle (which is cyclic), both diagonals have the same length d, and opposite sides are equal — say width w and height h. Ptolemy gives d · d = w · w + h · h, i.e. d² = w² + h². That's just the Pythagorean theorem, recovered as a special case.

It needs all four on one circle

Equality is special to cyclic quadrilaterals. For four points not all on a circle, the more general Ptolemy inequality holds instead: AC · BD ≤ AB · CD + AD · BC, with equality exactly when the points are concyclic and in this order. This tool keeps the points on the circle so you always see the equality.

What it's used for

Ptolemy used it to build his table of chords — an early trigonometry table — and it yields the angle-addition formulas for sine and cosine. It's also a workhorse in olympiad geometry.

A bit of history

It is named for Claudius Ptolemy (c. 100–170 CE), who presented it in the Almagest, his great astronomical treatise.