Tuesday, December 10, 2024

Ptolemy’s theorem for a quadrilateral inscribed in a circle

Computer sciencePtolemy's theorem for a quadrilateral inscribed in a circle




Draw a quadrilateral by pick four arbitrary points on a circle and connecting them cyclically.

Now multiply the lengths of the pairs of opposite sides. In the diagram below this means multiplying the lengths of the two horizontal-ish blue sides and the two vertical-ish orange sides.

quadrilateral with opposite sides colored

Ptolemy’s theorem says that the sum of the two products described above equals the product of the diagonals.

inscribed quadrilateral with diagonals

To put it in colorful terms, the product of the blue sides plus the product of the orange sides equals the product of the green diagonals.

The converse of Ptolemy’s theorem also holds. If the relationship above holds for a quadrilateral, then the quadrilateral can be inscribed in a circle.

Note that if the quadrilateral in Ptolemy’s theorem is a rectangle, then the theorem reduces to the Pythagorean theorem.

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