Editing Circle (section)

===Chord===
* Chords are equidistant from the centre of a circle if and only if they are equal in length.
* The [[perpendicular bisector]] of a chord passes through the centre of a circle; equivalent statements stemming from the uniqueness of the perpendicular bisector are:
** A perpendicular line from the centre of a circle bisects the chord.
** The [[line segment]] through the centre bisecting a chord is [[perpendicular]] to the chord.
* If a central angle and an [[inscribed angle]] of a circle are subtended by the same chord and on the same side of the chord, then the central angle is twice the inscribed angle.
* If two angles are inscribed on the same chord and on the same side of the chord, then they are equal.
* If two angles are inscribed on the same chord and on opposite sides of the chord, then they are [[supplementary angles|supplementary]].
** For a [[cyclic quadrilateral]], the [[exterior angle]] is equal to the interior opposite angle.
* An inscribed angle subtended by a diameter is a right angle (see [[Thales' theorem]]).
* The diameter is the longest chord of the circle.
** Among all the circles with a chord AB in common, the circle with minimal radius is the one with diameter AB.
* If the [[Intersecting chords theorem|intersection of any two chords]] divides one chord into lengths ''a'' and ''b'' and divides the other chord into lengths ''c'' and ''d'', then {{nowrap|''ab'' {{=}} ''cd''}}.
* If the intersection of any two perpendicular chords divides one chord into lengths ''a'' and ''b'' and divides the other chord into lengths ''c'' and ''d'', then {{nowrap|''a''<sup>2</sup> + ''b''<sup>2</sup> + ''c''<sup>2</sup> + ''d''<sup>2</sup>}} equals the square of the diameter.<ref>Posamentier and Salkind, ''Challenging Problems in Geometry'', Dover, 2nd edition, 1996: pp. 104–105, #4–23.</ref>
* The sum of the squared lengths of any two chords intersecting at right angles at a given point is the same as that of any other two perpendicular chords intersecting at the same point and is given by 8''r''<sup>2</sup> − 4''p''<sup>2</sup>, where ''r'' is the circle radius, and ''p'' is the distance from the centre point to the point of intersection.<ref>''[[College Mathematics Journal]]'' 29(4), September 1998, p. 331, problem 635.</ref>
* The distance from a point on the circle to a given chord times the diameter of the circle equals the product of the distances from the point to the ends of the chord.<ref>Johnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publ., 2007.</ref>{{rp|p.71}}