Editing Circle (section)

==Properties==
* The circle is the shape with the largest area for a given length of perimeter (see [[Isoperimetric inequality]]).
* The circle is a highly symmetric shape: every line through the centre forms a line of [[reflection symmetry]], and it has [[rotational symmetry]] around the centre for every angle. Its [[symmetry group]] is the [[orthogonal group]] O(2,''R''). The group of rotations alone is the [[circle group]] '''T'''.
* All circles are [[Similarity (geometry)|similar]].<ref>{{cite journal
 | last = Richeson | first = David
 | arxiv = 1303.0904
 | doi = 10.4169/college.math.j.46.3.162
 | issue = 3
 | journal = The College Mathematics Journal
 | mr = 3413900
 | pages = 162–171
 | title = Circular reasoning: who first proved that {{mvar|C}} divided by {{var|d}} is a constant?
 | volume = 46
 | year = 2015}}</ref>
** A circle circumference and radius are [[Proportionality (mathematics)|proportional]].
** The [[area (geometry)|area]] enclosed and the square of its radius are proportional.
** The constants of proportionality are 2{{pi}} and {{pi}} respectively.
* The circle that is centred at the origin with radius 1 is called the [[unit circle]].
** Thought of as a [[great circle]] of the [[unit sphere]], it becomes the [[Riemannian circle]].
* Through any three points, not all on the same line, there lies a unique circle. In Cartesian coordinates, it is possible to give explicit formulae for the coordinates of the centre of the circle and the radius in terms of the coordinates of the three given points. See [[circumcircle]].

===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}}

===Tangent===
* A line drawn perpendicular to a radius through the end point of the radius lying on the circle is a tangent to the circle.
* A line drawn perpendicular to a tangent through the point of contact with a circle passes through the centre of the circle.
* Two tangents can always be drawn to a circle from any point outside the circle, and these tangents are equal in length.
* If a tangent at ''A'' and a tangent at ''B'' intersect at the exterior point ''P'', then denoting the centre as ''O'', the angles ∠''BOA'' and ∠''BPA'' are supplementary.
* If ''AD'' is tangent to the circle at ''A'' and if ''AQ'' is a chord of the circle, then {{nowrap|∠''DAQ'' {{=}} {{sfrac|1|2}}arc(''AQ'')}}.

===Theorems===
[[Image:Secant-Secant Theorem.svg|thumb|right|Secant–secant theorem]]
{{See also|Power of a point}}
* The chord theorem states that if two chords, ''CD'' and ''EB'', intersect at ''A'', then {{nowrap|''AC'' × ''AD'' {{=}} ''AB'' × ''AE''}}.
* If two secants, ''AE'' and ''AD'', also cut the circle at ''B'' and ''C'' respectively, then {{nowrap|''AC'' × ''AD'' {{=}} ''AB'' × ''AE''}} (corollary of the chord theorem).
* {{anchor|Tangent-secant theorem}}A tangent can be considered a limiting case of a secant whose ends are coincident. If a tangent from an external point ''A'' meets the circle at ''F'' and a secant from the external point ''A'' meets the circle at ''C'' and ''D'' respectively, then {{nowrap|''AF''<sup>2</sup> {{=}} ''AC'' × ''AD''}} (tangent–secant theorem).
* The angle between a chord and the tangent at one of its endpoints is equal to one half the angle subtended at the centre of the circle, on the opposite side of the chord (tangent chord angle).
* If the angle subtended by the chord at the centre is 90[[Degree (angle)|°]], then {{nowrap|''ℓ'' {{=}} ''r'' √2}}, where ''ℓ'' is the length of the chord, and ''r'' is the radius of the circle.
* {{anchor|Secant-secant theorem}}If two secants are inscribed in the circle as shown at right, then the measurement of angle ''A'' is equal to one half the difference of the measurements of the enclosed arcs (<math>\overset{\frown}{DE}</math> and <math>\overset{\frown}{BC}</math>). That is, <math>2\angle{CAB} = \angle{DOE} - \angle{BOC}</math>, where ''O'' is the centre of the circle (secant–secant theorem).

===Inscribed angles===
{{See also|Inscribed angle theorem}}
[[Image:inscribed angle theorem.svg|thumb|200px|right|Inscribed-angle theorem]]
An inscribed angle (examples are the blue and green angles in the figure) is exactly half the corresponding [[central angle]] (red). Hence, all inscribed angles that subtend the same arc (pink) are equal. Angles inscribed on the arc (brown) are supplementary. In particular, every inscribed angle that subtends a diameter is a [[right angle]] (since the central angle is 180°).

{{clear}}

===Sagitta===
[[Image:circle Sagitta.svg|thumb|277px|right|The sagitta is the vertical segment.]]
The [[Sagitta (geometry)|sagitta]] (also known as the [[versine]]) is a line segment drawn perpendicular to a chord, between the midpoint of that chord and the arc of the circle.

Given the length ''y'' of a chord and the length ''x'' of the sagitta, the Pythagorean theorem can be used to calculate the radius of the unique circle that will fit around the two lines:
<math display="block">r = \frac{y^2}{8x} + \frac{x}{2}.</math>

Another proof of this result, which relies only on two chord properties given above, is as follows. Given a chord of length ''y'' and with sagitta of length ''x'', since the sagitta intersects the midpoint of the chord, we know that it is a part of a diameter of the circle. Since the diameter is twice the radius, the "missing" part of the diameter is ({{nowrap|2''r'' − ''x''}}) in length. Using the fact that one part of one chord times the other part is equal to the same product taken along a chord intersecting the first chord, we find that ({{nowrap|2''r'' − ''x'')''x'' {{=}} (''y'' / 2)<sup>2</sup>}}. Solving for ''r'', we find the required result.