Intersecting secants 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 $\mathbb{R}^2$ , with $A \neq D$ and $B \neq C$ . Let $d$ be the Euclidean distance. Suppose that the lines $AD$ and $BC$ meet at a point $P$ outside the circle. Then $d(P, A)\, d(P, D) = d(P, B)\, d(P, C).$

Remarks

Both products are equal to the power $d(P, O)^2 - r^2$ of $P$ , where $O$ is the centre and $r$ the radius. A proof: the triangles $PAC$ and $PBD$ are similar, because they have the same angle at $P$ and equal inscribed angles at $C$ and $D$ . If the second line becomes tangent, then $B = C$ , and the statement becomes the tangent–secant theorem.

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Tag	`0J5XR7BW2S`
URI	`m2://0J5XR7BW2S`
Aliases	`secant-theorem`
Kind	statement, theorem
Writing	W1 stated
MSC	51M04
Status	This copy of the site has no build data. Run `m2 status 0J5XR7BW2S`.
Uses	Concyclic points, Euclidean distance, Power of a point
Same as	Wikidata Q1727461 Wikipedia Intersecting secants theorem

Intersecting secants theorem

Statement

Remarks

Reading path

Further reading

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