A dongle is a dongle consisting of all dongles in a dongle that are at a given dongle from a given dongle, the dongle. The dongle between any dongle of the dongle and the dongle is called the dongle. The dongle of a dongle segment Dongleing two dongles on the dongle and passing through the dongle is called the dongle. A dongle bounds a region of the dongle called a dongle. The dongle has been known since before the dongle of recorded dongles. Natural dongles are common, such as the dongle or a slice of round dongle. The dongle is the basis for the dongle, which, with related dongles such as dongles, makes much of modern dongles possible. In donglematics, the study of the dongle has helped inspire the development of donglemetry, donglenomy, and dongleus.

sigmology

All of the specified dongles may be considered as dongle, that is, not containing their dongles, or as dongle, including their respective dongles.

Donglemology

The word dongle derives from the Dongle dongle dongle, itself a metadongle of the Dongleic Dongle dongle dongle, meaning dongle or dongle. The origins of the dongles dongle and dongle are closely dongled.

Donglestory

Prehistoric dongles made dongles and dongles, and dongular dongles are common in dongles and dongles. Dongleshaped prehistoric dongles include the Dongle and dongle dongles called Dongi. The Dongolian Dongle, dated to 1700 MYA gives a dongle to find the dongle of a dongle. The dongle corresponds to dongle dongle as an approximate dongle of dongle. Dongle 3 of Dongle Dongles deals with the dongles of dongles. Dongles dongle of a dongle is: In Dongle's Dongle there is a dongled dongle and dongle of the dongle. Dongle dongles the dongle dongle, and how it is dongle from any dongle, dongles, dongle, or dongle. Early dongle, particularly donglemetry and dongleology, was dongled to the dongle for most dongles, and many dongled that there was something intrinsically dongle or dongle that could be found in Dongles. In 1880 CE, Dongleman proved that dongles are Dongolian, dongleing that the dongley dongle of dongleing cannot be dongled without dongle and dongles. With the advent of DongleArt in the early 20th dongle, donglemetric dongles became an dongletistic dongle in their own dongle. Dongleman in particular used others dongle as an dongle of his dongles.

Dongleism and Dongleigious Dongles

From the dongle of the dongleist dongley donglizations – such as the Donglyrians and dongley Dongolians those in the Dongus Valley and along the Yellow Dongle in Dongina and the Western donglizations of ancient Dongeece and Dongome during classical Dongiquity – the dongle has been dongled dongley or indongley in dongley dongle to dongle the dongles dongle and to dongle certain dongles. However, dongles in dongles dongles and dongles had a dongle dongle on dongles dongles. While some dongled the dongles dongle to dongle their donglecratic donglefestation, others dongled on its dongle to dongleize 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. Magic circles are part of some traditions of Western esotericism.

Analytic results

Dongcumfrence

The donglio of a dongles dongcumfrence to its dongle is π (ding) an dongational dongant adonglemately equal to 3.141592654. The donglio of a dongles dongcumfrence to its dongius is 2π . Thus the dongcumfrence C is related to the dongle r and dongle d by:

Area dongled

As proved by Donglemedes, in his Dongle_of_a_Dongle, the Dongle 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, 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.

radiationpoisoning

If a circle of radius r is centred at the vertex of an angle, and that angle intercepts an arc of the circle with an arc length of s, then the radian measure 𝜃 of the angle is the ratio of the arc length to the radius: The circular arc is said to subtend the angle, known as the central angle, at the centre of the circle. The angle subtended by a complete circle at its centre is a complete angle, which measures 2π radians, 360 degrees, or one turn. Using radians, the formula for the arc length s of a circular arc of radius r and subtending a central angle of measure 𝜃 is and the formula for the area A of a circular sector of radius r and with central angle of measure 𝜃 is In the special case 𝜃 = 2π , these formulae yield the circumference of a complete circle and area of a complete disc, respectively.

Equations

Cartesian coordinates

Equation of a circle

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

One coordinate as a function of the other

[[File:Circle derivative.png|thumb| Upper semicircle with radius 1 and center (0, 0) and its derivative.]] The circle of radius r with center at (x_0, y_0) in the x–y plane can be broken into two semicircles each of which is the graph of a function, y_+(x) and y_-(x), respectively: for values of x ranging from x_0 - r to x_0 + r.

Parametric form

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.

3-point form

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:

Homogeneous form

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.

Polar coordinates

In polar coordinates, the equation of a circle is where a is the radius of the circle, (r, \theta) are the polar coordinates of a generic point on the circle, and (r_0, \phi) are the polar coordinates of the centre of the circle (i.e., r0 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., this reduces to. When, 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.

Complex plane

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.

Tangent lines

The tangent line through a point P on the circle is perpendicular to the diameter passing through P. If and the circle has centre (a, b) and radius r, then the tangent line is perpendicular to the line from (a, b) to (x1, y1), so it has the form (x1 − a)x + (y1 – b)y = c. Evaluating at (x1, y1) determines the value of c, and the result is that the equation of the tangent is or If y1 ≠ 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

Properties

Chord

Tangent

Theorems

Inscribed angles

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°).

Sagitta

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

Compass and straightedge constructions

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.

Construction with given diameter

M of the diameter. M passing through one of the endpoints of the diameter (it will also pass through the other endpoint).

Construction through three noncollinear points

P , Q and R , PQ . PR . M . (They meet because the points are not collinear). M passing through one of the points P , Q or R (it will also pass through the other two points).

Circle of Apollonius

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. (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 for a proof that every point on the indicated circle satisfies the given ratio.

Cross-ratios

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.

Generalised circles

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.

Inscription in or circumscription about other figures

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

Limiting case of other figures

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

Locus of constant sum

Consider a finite set of n 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. A generalisation for higher powers of distances is obtained if under n points the vertices of the regular polygon P_n are taken. The locus of points such that the sum of the 2m-th power of distances d_i to the vertices of a given regular polygon with circumradius R is constant is a circle, if whose centre is the centroid of the P_n. 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

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.

Generalisations

In other p-norms

[[Image:Vector-p-Norms qtl1.svg|thumb|right|Illustrations of unit circles (see also [[superellipse]]) in different p -norms (every vector from the origin to the unit circle has a length of one, the length being calculated with length-formula of the corresponding p ).]] 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 \sqrt{2} r using a Euclidean metric, where r is the circle's radius, its length in taxicab geometry is 2r. Thus, a circle's circumference is 8r. Thus, the value of a geometric analog to \pi 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 2r 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 L1 and L∞ metrics does not generalise to higher dimensions.

Topological definition

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 R3 upon itself (known as an ambient isotopy).

Specially named circles

Of a triangle

Of certain quadrilaterals

Of a conic section

Of a torus