Intersecting chords theorem

Builds on Concyclic points Euclidean distance Power of a point

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 chords $AC$ and $BD$ meet at a point $S$ that is inside both segments. Then $d(A, S)\, d(S, C) = d(B, S)\, d(S, D).$

Remarks

Both products are equal to $r^2 - d(S, O)^2$ , where $O$ is the centre and $r$ the radius. This is minus the power of $S$ . A proof: the triangles $ASD$ and $BSC$ are similar, by the inscribed angle theorem. The theorem is Proposition 35 of Book III of Euclid's Elements.

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Tag	`0YQQYET8YQ`
URI	`m2://0YQQYET8YQ`
Aliases	`chord-theorem`
Kind	statement, theorem
Writing	W1 stated
MSC	51M04
Status	This copy of the site has no build data. Run `m2 status 0YQQYET8YQ`.
Uses	Concyclic points, Euclidean distance, Power of a point
Same as	Wikidata Q2266397 Wikipedia Intersecting chords theorem

Intersecting chords theorem

Statement

Remarks

Reading path

Further reading

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