Ptolemy's theorem

Builds on Cyclic quadrilateral Concyclic points Euclidean distance

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.

Reading path

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Tag	`058YS05YP5`
URI	`m2://058YS05YP5`
Kind	statement, theorem
Writing	W1 stated
MSC	51M04
Status	This copy of the site has no build data. Run `m2 status 058YS05YP5`.
Uses	Cyclic quadrilateral, Concyclic points, Euclidean distance
Same as	Wikidata Q459547 Wikipedia Ptolemy's theorem

Ptolemy's theorem

Statement

Remarks

History

Reading path

Further reading

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