Circle | |
---|---|

Type | Conic section |

Symmetry group | O(2) |

Area | πR^{2} |

Perimeter | C = 2πR |

Geometry |
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Geometers |

A **circle** is a shape consisting of all points in a plane that are at a given distance from a given point, thecentre. The distance between any point of the circle and the centre is called the radius.

- Terminology
- Etymology
- History
- Symbolism and religious use
- Analytic results
- Circumference
- Area enclosed
- Equations
- Tangent lines
- Properties
- Chord
- Tangent
- Theorems
- Inscribed angles
- Sagitta
- Compass and straightedge constructions
- Construction with given diameter
- Construction through three noncollinear points
- Circle of Apollonius
- Cross-ratios
- Generalised circles
- Inscription in or circumscription about other figures
- Limiting case of other figures
- Locus of constant sum
- Squaring the circle
- Generalizations
- In other p-norms
- Topological definition
- Specially named circles
- Of a triangle
- Of certain quadrilaterals
- Of a conic section
- Of a torus
- See also
- References
- Further reading
- External links

The circle has been known since before the beginning of recorded history. Natural circles are common, such as the full moon or a slice of round fruit. The circle is the basis for the wheel, which, with related inventions such as gears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus.

- Annulus: a ring-shaped object, the region bounded by two concentric circles.
- Arc: any connected part of a circle. Specifying two end points of an arc and a centre allows for two arcs that together make up a full circle.
- Centre: the point equidistant from all points on the circle.
- Chord: a line segment whose endpoints lie on the circle, thus dividing a circle into two segments.
- Circumference: the length of one circuit along the circle, or the distance around the circle.
- Diameter: a line segment whose endpoints lie on the circle and that passes through the centre; or the length of such a line segment. This is the largest distance between any two points on the circle. It is a special case of a chord, namely the longest chord for a given circle, and its length is twice the length of a radius.
- Disc: the region of the plane bounded by a circle. In strict mathematical usage, a circle is only the boundary of the disc, while in everyday the terms "circle" and "disc" may be used interchangeably.
- Lens: the region common to (the intersection of) two overlapping discs.
- Radius: a line segment joining the centre of a circle with any single point on the circle itself; or the length of such a segment, which is half (the length of) a diameter. Usually, the radius is denoted and required to be a positive number. A circle with is a degenerate case consisting of a single point.
- Sector: a region bounded by two radii of equal length with a common centre and either of the two possible arcs, determined by this centre and the endpoints of the radii.
- Segment: a region bounded by a chord and one of the arcs connecting the chord's endpoints. The length of the chord imposes a lower boundary on the diameter of possible arcs. Sometimes the term
*segment*is used only for regions not containing the centre of the circle to which their arc belongs to. - Secant: an extended chord, a coplanar straight line, intersecting a circle in two points.
- Semicircle: one of the two possible arcs determined by the endpoints of a diameter, taking its midpoint as centre. In non-technical common usage it may mean the interior of the two-dimensional region bounded by a diameter and one of its arcs, that is technically called a half-disc. A half-disc is a special case of a segment, namely the largest one.
- Tangent: a coplanar straight line that has one single point in common with a circle ("touches the circle at this point").

All of the specified regions may be considered as *open*, that is, not containing their boundaries, or as *closed*, including their respective boundaries.

The word *circle* derives from the Greek κίρκος/κύκλος (*kirkos/kuklos*), itself a metathesis of the Homeric Greek κρίκος (*krikos*), meaning "hoop" or "ring".^{ [1] } The origins of the words * circus * and * circuit * are closely related.

Prehistoric people made stone circles and timber circles, and circular elements are common in petroglyphs and cave paintings.^{ [2] } Disc-shaped prehistoric artifacts include the Nebra sky disc and jade discs called Bi.

The Egyptian Rhind papyrus, dated to 1700 BCE, gives a method to find the area of a circle. The result corresponds to 256/81 (3.16049...) as an approximate value of π.^{ [3] }

Book 3 of Euclid's *Elements* deals with the properties of circles. Euclid's definition of a circle is:

A circle is a plane figure bounded by one curved line, and such that all straight lines drawn from a certain point within it to the bounding line, are equal. The bounding line is called its circumference and the point, its centre.

In Plato's Seventh Letter there is a detailed definition and explanation of the circle. Plato explains the perfect circle, and how it is different from any drawing, words, definition or explanation. Early science, particularly geometry and astrology and astronomy, was connected to the divine for most medieval scholars, and many believed that there was something intrinsically "divine" or "perfect" that could be found in circles.^{ [5] }^{ [6] }

In 1880 CE, Ferdinand von Lindemann proved that π is transcendental, proving that the millennia-old problem of squaring the circle cannot be performed with straightedge and compass.^{ [7] }

With the advent of abstract art in the early 20th century, geometric objects became an artistic subject in their own right. Wassily Kandinsky in particular often used circles as an element of his compositions.^{ [8] }^{ [9] }

From the time of the earliest known civilisations – such as the Assyrians and ancient Egyptians, those in the Indus Valley and along the Yellow River in China, and the Western civilisations of ancient Greece and Rome during classical Antiquity – the circle has been used directly or indirectly in visual art to convey the artist's message and to express certain ideas. However, differences in worldview (beliefs and culture) had a great impact on artists' perceptions. While some emphasised the circle's perimeter to demonstrate their democratic manifestation, others focused on its centre to symbolise the concept of cosmic unity. In mystical doctrines, the circle mainly symbolises the infinite and cyclical nature of existence, but in religious traditions it represents heavenly bodies and divine spirits.

The circle signifies many sacred and spiritual concepts, including unity, infinity, wholeness, the universe, divinity, balance, stability and perfection, among others. Such concepts have been conveyed in cultures worldwide through the use of symbols, for example, a compass, a halo, the vesica piscis and its derivatives (fish, eye, aureole, mandorla, etc.), the ouroboros, the Dharma wheel, a rainbow, mandalas, rose windows and so forth.^{ [10] } Magic circles are part of some traditions of Western esotericism.

The ratio of a circle's circumference to its diameter is π (pi), an irrational constant approximately equal to 3.141592654. Thus the circumference *C* is related to the radius *r* and diameter *d* by:

As proved by Archimedes, in his Measurement of a Circle, the area enclosed by a circle is equal to that of a triangle whose base has the length of the circle's circumference and whose height equals the circle's radius,^{ [11] } which comes to π multiplied by the radius squared:

Equivalently, denoting diameter by *d*,

that is, approximately 79% of the circumscribing square (whose side is of length *d*).

The circle is the plane curve enclosing the maximum area for a given arc length. This relates the circle to a problem in the calculus of variations, namely the isoperimetric inequality.

In an *x*–*y* Cartesian coordinate system, the circle with centre coordinates (*a*, *b*) and radius *r* is the set of all points (*x*, *y*) such that

This equation, known as the *equation of the circle*, follows from the Pythagorean theorem applied to any point on the circle: as shown in the adjacent diagram, the radius is the hypotenuse of a right-angled triangle whose other sides are of length |*x* − *a*| and |*y* − *b*|. If the circle is centred at the origin (0, 0), then the equation simplifies to

The equation can be written in parametric form using the trigonometric functions sine and cosine as

where *t* is a parametric variable in the range 0 to 2π, interpreted geometrically as the angle that the ray from (*a*, *b*) to (*x*, *y*) makes with the positive *x* axis.

An alternative parametrisation of the circle is

In this parameterisation, the ratio of *t* to *r* can be interpreted geometrically as the stereographic projection of the line passing through the centre parallel to the *x* axis (see Tangent half-angle substitution). However, this parameterisation works only if *t* is made to range not only through all reals but also to a point at infinity; otherwise, the leftmost point of the circle would be omitted.

The equation of the circle determined by three points not on a line is obtained by a conversion of the *3-point form of a circle equation*:

In homogeneous coordinates, each conic section with the equation of a circle has the form

It can be proven that a conic section is a circle exactly when it contains (when extended to the complex projective plane) the points *I*(1: *i*: 0) and *J*(1: −*i*: 0). These points are called the circular points at infinity.

In polar coordinates, the equation of a circle is

where *a* is the radius of the circle, are the polar coordinates of a generic point on the circle, and are the polar coordinates of the centre of the circle (i.e., *r*_{0} is the distance from the origin to the centre of the circle, and *φ* is the anticlockwise angle from the positive *x* axis to the line connecting the origin to the centre of the circle). For a circle centred on the origin, i.e. *r*_{0} = 0, this reduces to *r* = *a*. When *r*_{0} = *a*, or when the origin lies on the circle, the equation becomes

In the general case, the equation can be solved for *r*, giving

Without the ± sign, the equation would in some cases describe only half a circle.

In the complex plane, a circle with a centre at *c* and radius *r* has the equation

In parametric form, this can be written as

The slightly generalised equation

for real *p*, *q* and complex *g* is sometimes called a generalised circle. This becomes the above equation for a circle with , since . Not all generalised circles are actually circles: a generalised circle is either a (true) circle or a line.

The tangent line through a point *P* on the circle is perpendicular to the diameter passing through *P*. If P = (*x*_{1}, *y*_{1}) and the circle has centre (*a*, *b*) and radius *r*, then the tangent line is perpendicular to the line from (*a*, *b*) to (*x*_{1}, *y*_{1}), so it has the form (*x*_{1} − *a*)*x* + (*y*_{1} – *b*)*y* = *c*. Evaluating at (*x*_{1}, *y*_{1}) determines the value of *c*, and the result is that the equation of the tangent is

or

If *y*_{1} ≠ *b*, then the slope of this line is

This can also be found using implicit differentiation.

When the centre of the circle is at the origin, then the equation of the tangent line becomes

and its slope is

- 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 similar.
^{ [12] }- A circle circumference and radius are proportional.
- The area enclosed and the square of its radius are proportional.
- The constants of proportionality are 2π and π 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.

- 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.
- 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 intersection of any two chords divides one chord into lengths
*a*and*b*and divides the other chord into lengths*c*and*d*, then*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*a*^{2}+*b*^{2}+*c*^{2}+*d*^{2}equals the square of the diameter.^{ [13] } - 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*^{2}− 4*p*^{2}, where*r*is the circle radius, and*p*is the distance from the centre point to the point of intersection.^{ [14] } - 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.
^{ [15] }^{: p.71 }

- 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 ∠*DAQ*= 1/2arc(*AQ*).

- The chord theorem states that if two chords,
*CD*and*EB*, intersect at*A*, then*AC*×*AD*=*AB*×*AE*. - If two secants,
*AE*and*AD*, also cut the circle at*B*and*C*respectively, then*AC*×*AD*=*AB*×*AE*(corollary of the chord 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*AF*^{2}=*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°, then
*ℓ*=*r*√2, where*ℓ*is the length of the chord, and*r*is the radius of the circle. - 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 ( and ). That is, , where*O*is the centre of the circle (secant–secant 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°).

The 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:

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 (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 (2*r* − *x*)*x* = (*y* / 2)^{2}. Solving for *r*, we find the required result.

There are many compass-and-straightedge constructions resulting in circles.

The simplest and most basic is the construction given the centre of the circle and a point on the circle. Place the fixed leg of the compass on the centre point, the movable leg on the point on the circle and rotate the compass.

- Construct the midpoint
**M**of the diameter. - Construct the circle with centre
**M**passing through one of the endpoints of the diameter (it will also pass through the other endpoint).

- Name the points
**P**,**Q**and**R**, - Construct the perpendicular bisector of the segment
**PQ**. - Construct the perpendicular bisector of the segment
**PR**. - Label the point of intersection of these two perpendicular bisectors
**M**. (They meet because the points are not collinear). - Construct the circle with centre
**M**passing through one of the points**P**,**Q**or**R**(it will also pass through the other two points).

Apollonius of Perga showed that a circle may also be defined as the set of points in a plane having a constant *ratio* (other than 1) of distances to two fixed foci, *A* and *B*.^{ [16] }^{ [17] } (The set of points where the distances are equal is the perpendicular bisector of segment *AB*, a line.) That circle is sometimes said to be drawn *about* two points.

The proof is in two parts. First, one must prove that, given two foci *A* and *B* and a ratio of distances, any point *P* satisfying the ratio of distances must fall on a particular circle. Let *C* be another point, also satisfying the ratio and lying on segment *AB*. By the angle bisector theorem the line segment *PC* will bisect the interior angle *APB*, since the segments are similar:

Analogously, a line segment *PD* through some point *D* on *AB* extended bisects the corresponding exterior angle *BPQ* where *Q* is on *AP* extended. Since the interior and exterior angles sum to 180 degrees, the angle *CPD* is exactly 90 degrees; that is, a right angle. The set of points *P* such that angle *CPD* is a right angle forms a circle, of which *CD* is a diameter.

Second, see^{ [18] }^{: 15 } for a proof that every point on the indicated circle satisfies the given ratio.

A closely related property of circles involves the geometry of the cross-ratio of points in the complex plane. If *A*, *B*, and *C* are as above, then the circle of Apollonius for these three points is the collection of points *P* for which the absolute value of the cross-ratio is equal to one:

Stated another way, *P* is a point on the circle of Apollonius if and only if the cross-ratio [*A*, *B*; *C*, *P*] is on the unit circle in the complex plane.

If *C* is the midpoint of the segment *AB*, then the collection of points *P* satisfying the Apollonius condition

is not a circle, but rather a line.

Thus, if *A*, *B*, and *C* are given distinct points in the plane, then the locus of points *P* satisfying the above equation is called a "generalised circle." It may either be a true circle or a line. In this sense a line is a generalised circle of infinite radius.

In every triangle a unique circle, called the incircle, can be inscribed such that it is tangent to each of the three sides of the triangle.^{ [19] }

About every triangle a unique circle, called the circumcircle, can be circumscribed such that it goes through each of the triangle's three vertices.^{ [20] }

A tangential polygon, such as a tangential quadrilateral, is any convex polygon within which a circle can be inscribed that is tangent to each side of the polygon.^{ [21] } Every regular polygon and every triangle is a tangential polygon.

A cyclic polygon is any convex polygon about which a circle can be circumscribed, passing through each vertex. A well-studied example is the cyclic quadrilateral. Every regular polygon and every triangle is a cyclic polygon. A polygon that is both cyclic and tangential is called a bicentric polygon.

A hypocycloid is a curve that is inscribed in a given circle by tracing a fixed point on a smaller circle that rolls within and tangent to the given circle.

The circle can be viewed as a limiting case of various other figures:

- The series of regular polygons with
*n*sides has the circle as its limit as*n*approaches infinity. This fact was applied by Archimedes to approximate π. - A Cartesian oval is a set of points such that a weighted sum of the distances from any of its points to two fixed points (foci) is a constant. An ellipse is the case in which the weights are equal. A circle is an ellipse with an eccentricity of zero, meaning that the two foci coincide with each other as the centre of the circle. A circle is also a different special case of a Cartesian oval in which one of the weights is zero.
- A superellipse has an equation of the form for positive
*a*,*b*, and*n*. A supercircle has*b*=*a*. A circle is the special case of a supercircle in which*n*= 2. - A Cassini oval is a set of points such that the product of the distances from any of its points to two fixed points is a constant. When the two fixed points coincide, a circle results.
- A curve of constant width is a figure whose width, defined as the perpendicular distance between two distinct parallel lines each intersecting its boundary in a single point, is the same regardless of the direction of those two parallel lines. The circle is the simplest example of this type of figure.

Consider a finite set of points in the plane. The locus of points such that the sum of the squares of the distances to the given points is constant is a circle, whose centre is at the centroid of the given points.^{ [22] } A generalization for higher powers of distances is obtained if under points the vertices of the regular polygon are taken.^{ [23] } The locus of points such that the sum of the -th power of distances to the vertices of a given regular polygon with circumradius is constant is a circle, if

whose centre is the centroid of the .

In the case of the equilateral triangle, the loci of the constant sums of the second and fourth powers are circles, whereas for the square, the loci are circles for the constant sums of the second, fourth, and sixth powers. For the regular pentagon the constant sum of the eighth powers of the distances will be added and so forth.

Squaring the circle is the problem, proposed by ancient geometers, of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge.

In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem, which proves that pi (π) is a transcendental number, rather than an algebraic irrational number; that is, it is not the root of any polynomial with rational coefficients. Despite the impossibility, this topic continues to be of interest for pseudomath enthusiasts.

Defining a circle as the set of points with a fixed distance from a point, different shapes can be considered circles under different definitions of distance. In *p*-norm, distance is determined by

In Euclidean geometry, *p* = 2, giving the familiar

In taxicab geometry, *p* = 1. Taxicab circles are squares with sides oriented at a 45° angle to the coordinate axes. While each side would have length using a Euclidean metric, where *r* is the circle's radius, its length in taxicab geometry is 2*r*. Thus, a circle's circumference is 8*r*. Thus, the value of a geometric analog to is 4 in this geometry. The formula for the unit circle in taxicab geometry is in Cartesian coordinates and

in polar coordinates.

A circle of radius 1 (using this distance) is the von Neumann neighborhood of its centre.

A circle of radius *r* for the Chebyshev distance (*L*_{∞} metric) on a plane is also a square with side length 2*r* parallel to the coordinate axes, so planar Chebyshev distance can be viewed as equivalent by rotation and scaling to planar taxicab distance. However, this equivalence between *L*_{1} and *L*_{∞} metrics does not generalize to higher dimensions.

The circle is the one-dimensional hypersphere (the 1-sphere).

In topology, a circle is not limited to the geometric concept, but to all of its homeomorphisms. Two topological circles are equivalent if one can be transformed into the other via a deformation of **R**^{3} upon itself (known as an ambient isotopy).^{ [24] }

## Of a triangle | ## Of certain quadrilaterals- Eight-point circle of an orthodiagonal quadrilateral
## Of a conic section## Of a torus |

In mathematics, an **ellipse** is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity , a number ranging from to .

In mathematics, a **hyperbola** is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola.

In mathematics, a **parabola** is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.

A **sphere** is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. Formally, a sphere is the set of points that are all at the same distance *r* from a given point in three-dimensional space. That given point is the center of the sphere, and *r* is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians.

In geometry, two geometric objects are **perpendicular** if their intersection forms right angles at the point of intersection called a *foot*. The condition of **perpendicularity** may be represented graphically using the *perpendicular symbol*, ⟂. Perpendicular intersections can happen between two lines, between a line and a plane, and between two planes.

In geometry, the **incircle** or **inscribed circle** of a triangle is the largest circle that can be contained in the triangle; it touches the three sides. The center of the incircle is a triangle center called the triangle's incenter.

In geometry, **inversive geometry** is the study of *inversion*, a transformation of the Euclidean plane that maps circles or lines to other circles or lines and that preserves the angles between crossing curves. Many difficult problems in geometry become much more tractable when an inversion is applied. Inversion seems to have been discovered by a number of people contemporaneously, including Steiner (1824), Quetelet (1825), Bellavitis (1836), Stubbs and Ingram (1842–3) and Kelvin (1845).

In geometry, the **incenter** of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisectors of the triangle cross, as the point equidistant from the triangle's sides, as the junction point of the medial axis and innermost point of the grassfire transform of the triangle, and as the center point of the inscribed circle of the triangle.

In geometry, a set of points are said to be **concyclic** if they lie on a common circle. A polygon whose vertices are concyclic is called a **cyclic polygon**, and the circle is called its *circumscribing circle* or *circumcircle*. All concyclic points are equidistant from the center of the circle.

In mathematics, a **semicircle** is a one-dimensional locus of points that forms half of a circle. It is a circular arc that measures 180°. It has only one line of symmetry.

In mathematics, a **pedal curve** of a given curve results from the orthogonal projection of a fixed point on the tangent lines of this curve. More precisely, for a plane curve *C* and a given fixed *pedal point**P*, the **pedal curve** of *C* is the locus of points *X* so that the line *PX* is perpendicular to a tangent *T* to the curve passing through the point *X*. Conversely, at any point *R* on the curve *C*, let *T* be the tangent line at that point *R*; then there is a unique point *X* on the tangent *T* which forms with the pedal point *P* a line perpendicular to the tangent *T* – the pedal curve is the set of such points *X*, called the *foot* of the perpendicular to the tangent *T* from the fixed point *P*, as the variable point *R* ranges over the curve *C*.

In hyperbolic geometry, two lines are said to be **ultraparallel** if they do not intersect and are not limiting parallel.

In geometry, the **circumscribed circle** or **circumcircle** of a triangle is a circle that passes through all three vertices. The center of this circle is called the **circumcenter** of the triangle, and its radius is called the **circumradius**. The circumcenter is the point of intersection between the three perpendicular bisectors of the triangle's sides, and is a triangle center.

In geometry, the **Beltrami–Klein model**, also called the **projective model**, **Klein disk model**, and the **Cayley–Klein model**, is a model of hyperbolic geometry in which points are represented by the points in the interior of the unit disk and lines are represented by the chords, straight line segments with ideal endpoints on the boundary sphere.

A **circular arc** is the arc of a circle between a pair of distinct points. If the two points are not directly opposite each other, one of these arcs, the **minor arc**, subtends an angle at the center of the circle that is less than π radians ; and the other arc, the **major arc**, subtends an angle greater than π radians. The arc of a circle is defined as the part or segment of the circumference of a circle. A straight line that connects the two ends of the arc is known as a *chord* of a circle. If the length of an arc is exactly half of the circle, it is known as a *semicircular arc*.

In geometry, a **homothetic center** is a point from which at least two geometrically similar figures can be seen as a dilation or contraction of one another. If the center is **external**, the two figures are directly similar to one another; their angles have the same rotational sense. If the center is **internal**, the two figures are scaled mirror images of one another; their angles have the opposite sense.

In Euclidean plane geometry, a **tangent line to a circle** is a line that touches the circle at exactly one point, never entering the circle's interior. Tangent lines to circles form the subject of several theorems, and play an important role in many geometrical constructions and proofs. Since the tangent line to a circle at a point P is perpendicular to the radius to that point, theorems involving tangent lines often involve radial lines and orthogonal circles.

In Euclidean geometry, a **bicentric quadrilateral** is a convex quadrilateral that has both an incircle and a circumcircle. The radii and centers of these circles are called *inradius* and *circumradius*, and *incenter* and *circumcenter* respectively. From the definition it follows that bicentric quadrilaterals have all the properties of both tangential quadrilaterals and cyclic quadrilaterals. Other names for these quadrilaterals are **chord-tangent quadrilateral** and **inscribed and circumscribed quadrilateral**. It has also rarely been called a *double circle quadrilateral* and *double scribed quadrilateral*.

In Euclidean geometry, an **orthodiagonal quadrilateral** is a quadrilateral in which the diagonals cross at right angles. In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal (perpendicular) to each other.

In geometry, the **Poincaré disk model**, also called the **conformal disk model**, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk that are orthogonal to the unit circle or diameters of the unit circle.

- ↑ krikos Archived 2013-11-06 at the Wayback Machine , Henry George Liddell, Robert Scott,
*A Greek-English Lexicon*, on Perseus - ↑ Simek, Jan F.; Cressler, Alan; Herrmann, Nicholas P.; Sherwood, Sarah C. (1 June 2013). "Sacred landscapes of the south-eastern USA: prehistoric rock and cave art in Tennessee".
*Antiquity*.**87**(336): 430–446. doi:10.1017/S0003598X00049048. ISSN 0003-598X. S2CID 130296519. - ↑ Chronology for 30000 BC to 500 BC Archived 2008-03-22 at the Wayback Machine . History.mcs.st-andrews.ac.uk. Retrieved on 2012-05-03.
- ↑ OL 7227282M
- ↑ Arthur Koestler,
*The Sleepwalkers: A History of Man's Changing Vision of the Universe*(1959) - ↑ Proclus,
*The Six Books of Proclus, the Platonic Successor, on the Theology of Plato*Archived 2017-01-23 at the Wayback Machine Tr. Thomas Taylor (1816) Vol. 2, Ch. 2, "Of Plato" - ↑ Squaring the circle Archived 2008-06-24 at the Wayback Machine . History.mcs.st-andrews.ac.uk. Retrieved on 2012-05-03.
- ↑ "Circles in a Circle".
*Philadelphia Museum of Art*. Retrieved 28 December 2023. - ↑ Lesso, Rosie (15 June 2022). "Why Did Wassily Kandinsky Paint Circles?".
*TheCollector*. Retrieved 28 December 2023. - ↑ Abdullahi, Yahya (29 October 2019). "The Circle from East to West". In Charnier, Jean-François (ed.).
*The Louvre Abu Dhabi: A World Vision of Art*. Rizzoli International Publications, Incorporated. ISBN 9782370741004. - ↑ Katz, Victor J. (1998).
*A History of Mathematics / An Introduction*(2nd ed.). Addison Wesley Longman. p. 108. ISBN 978-0-321-01618-8. - ↑ Richeson, David (2015). "Circular reasoning: who first proved that C divided by
`d`is a constant?".*The College Mathematics Journal*.**46**(3): 162–171. arXiv: 1303.0904 . doi:10.4169/college.math.j.46.3.162. MR 3413900. - ↑ Posamentier and Salkind,
*Challenging Problems in Geometry*, Dover, 2nd edition, 1996: pp. 104–105, #4–23. - ↑
*College Mathematics Journal*29(4), September 1998, p. 331, problem 635. - ↑ Johnson, Roger A.,
*Advanced Euclidean Geometry*, Dover Publ., 2007. - ↑ Harkness, James (1898). "Introduction to the theory of analytic functions".
*Nature*.**59**(1530): 30. Bibcode:1899Natur..59..386B. doi:10.1038/059386a0. S2CID 4030420. Archived from the original on 7 October 2008. - ↑ Ogilvy, C. Stanley,
*Excursions in Geometry*, Dover, 1969, 14–17. - ↑ Altshiller-Court, Nathan,
*College Geometry*, Dover, 2007 (orig. 1952). - ↑ Incircle – from Wolfram MathWorld Archived 2012-01-21 at the Wayback Machine . Mathworld.wolfram.com (2012-04-26). Retrieved on 2012-05-03.
- ↑ Circumcircle – from Wolfram MathWorld Archived 2012-01-20 at the Wayback Machine . Mathworld.wolfram.com (2012-04-26). Retrieved on 2012-05-03.
- ↑ Tangential Polygon – from Wolfram MathWorld Archived 2013-09-03 at the Wayback Machine . Mathworld.wolfram.com (2012-04-26). Retrieved on 2012-05-03.
- ↑ Apostol, Tom; Mnatsakanian, Mamikon (2003). "Sums of squares of distances in m-space".
*American Mathematical Monthly*.**110**(6): 516–526. doi:10.1080/00029890.2003.11919989. S2CID 12641658. - ↑ Meskhishvili, Mamuka (2020). "Cyclic Averages of Regular Polygons and Platonic Solids".
*Communications in Mathematics and Applications*.**11**: 335–355. arXiv: 2010.12340 . doi:10.26713/cma.v11i3.1420 (inactive 31 January 2024). Archived from the original on 22 April 2021. Retrieved 17 May 2021.`{{cite journal}}`

: CS1 maint: DOI inactive as of January 2024 (link) - ↑ Gamelin, Theodore (1999).
*Introduction to topology*. Mineola, N.Y: Dover Publications. ISBN 0486406806.

- Pedoe, Dan (1988).
*Geometry: a comprehensive course*. Dover. ISBN 9780486658124. - "Circle" in The MacTutor History of Mathematics archive Archived 7 April 2019 at the Wayback Machine

Wikiquote has quotations related to ** Circles **.

- "Circle".
*Encyclopedia of Mathematics*. EMS Press. 2001 [1994]. - Circle at PlanetMath .
- Weisstein, Eric W. "Circle".
*MathWorld*. - "Interactive Java applets".
for the properties of and elementary constructions involving circles

- "Interactive Standard Form Equation of Circle".
Click and drag points to see standard form equation in action

- "Munching on Circles". Cut-the-Knot.

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