Chords in a Circle

Chords

Definition: Chord

A chord in a circle is any line segment whose endpoints are points of the circle.

Properties

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 radius $r$ and the distance $l$ between $O$ as follows:

$$|AS|\cdot |SC|=|BS|\cdot |BD|=r^2-l^2$$

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.

PROOF

TODO

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.

$$|AB|=|CD|⟺|\stackrel{⌢}{\mathrm{AB}}|=|\stackrel{⌢}{\mathrm{CD}}|$$

PROOF

TODO

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 $CD$.

$$|AB|=|CD|⟺d(O;AB)=d(O;CD)$$

PROOF

TODO

Theorem: Parallel Chords $⟹$ Equal Arcs

The arcs subtended between two parallel chords $AB$ and $CD$ in a circle always have equal length.

$$AB\parallel CD⟹|\stackrel{⌢}{\mathrm{AC}}|=|\stackrel{⌢}{\mathrm{BD}}|$$

PROOF

TODO

Theorem: Diameter and Chord Perpendicularity

Let $AB$ be a non-diameter chord in a circle $k(O;r)$.

A diameter $d$ of $k$ is perpendicular to $AB$ if and only if it splits $AB$ in half.

$$d\perp AB⟺|AS|=|BS|$$

PROOF

TODO