Chords#

Intersecting Chords Theorem

If \(AC\) and \(BD\) are [[Chords]] in a [[Circle]] \(k(O; r)\) and they intersect at \(S\), then \(S\) divides each chord such that the product of the lengths of the two segments of one chord is equal to the product of the lengths of the segments of the other chord. Moreover, this product is given by the [[Circle#Circle|radius]] \(r\) and the distance \(l\) between \(O\) as follows:

![[res/Intersecting Chords Theorem.svg]]

The converse is also true - if two [[Line Segments]] \(AC\) and \(BD\) intersect at a point \(S\) which divides each segment such that \(|AS| \cdot |SC| = |BS| \cdot |BD|\), then \(AC\) and \(BD\) are [[Chords]] in a [[Circle#Circle|circle]].

Theorem: Subtended Arcs and Chords

Two [[Chords]] \(AB\) and \(CD\) in a [[Circle]] \(k(O; r)\) have equal length if and only if the arcs they subtend have equal length.

![[res/Equal Chords iff Equal Arcs.svg]]

Theorem: Chords and Distances

Two [[Chords]] \(AB\) and \(CD\) in a [[Circle]] \(k(O; r)\) have equal length if and only if the distance between \(O\) and \(AB\) is is the same as the distance between \(O\) and \(CD\).

![[res/Equal Chords iff Equal Distances.svg]]

Theorem: Parallel Chords \(\implies\) Equal Arcs

The arcs subtended between two [[TODO|parallel]] [[Chords]] \(AB\) and \(CD\) in a [[Circle#Circle|circle]] always have equal length.

![[res/Parallel Chords imply Equal Arcs.svg]]

Theorem: Diameter and Chord Perpendicularity

Let \(AB\) be a non-[[Circle#Circle|diameter]] [[Chords|chord]] in a [[Circle#Circle|circle]] \(k(O;r)\).

A [[Circle#Circle|diameter]] \(d\) of \(k\) is [[TODO|perpendicular]] to \(AB\) if and only if it splits \(AB\) in half.

![[res/Diameter and Chord Perpendicularity.svg]]