circle
American Heritage Dictionary:
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cir·cle
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n.
- A plane curve everywhere equidistant from a given fixed point, the center.
- A planar region bounded by a circle.
- Something, such as a ring, shaped like such a plane curve.
- A circular course, circuit, or orbit: a satellite's circle around the earth.
- A traffic circle.
- A curved section or tier of seats in a theater.
- A series or process that finishes at its starting point or continuously repeats itself; a cycle.
- A group of people sharing an interest, activity, or achievement: well-known in artistic circles.
- A territorial or administrative division, especially of a province, in some European countries.
- A sphere of influence or interest; domain.
- Logic. A vicious circle.
v., -cled, -cling, -cles. v.tr.
- To make or form a circle around; enclose. See synonyms at surround.
- To move in a circle around.
To move in a circle. See synonyms at turn.
idiom:
circle the wagons
- To take a defensive position; become defensive.
[Middle English cercle, from Old French, from Latin circulus, diminutive of circus, circle, from Greek kirkos, krikos.]
circler cir'cler (-klər) n.Related Videos:
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circle
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Geometrical curve, one of the conic sections, consisting of the set of all points the same distance (the radius) from a given point (the centre). A line connecting any two points on a circle is called a chord, and a chord passing through the centre is called a diameter. The distance around a circle (the circumference) equals the length of a diameter multiplied by ( pi). The area of a circle is the square of the radius multiplied by . An arc consists of any part of a circle encompassed by an angle with its vertex at the centre (central angle). Its length is in the same proportion to the circumference as the central angle is to a full revolution.
For more information on circle, visit Britannica.com.
The curve that is the locus of points in a plane with equal distance (radius) from a fixed point (center). In elementary mathematics, circle often refers to the finite portion of the plane bounded by a curve (circumference) all points of which are equidistant from a fixed point of the plane, that is, a circular disk. Circles are conic sections and are defined analytically by certain second-degree equations in cartesian coordinates. The ancient Greeks formulated the problem of “squaring the circle,” that is, to construct, with compasses and unmarked straightedge only, a square whose area is equal to that of a given circle. It was not until 1882 that this was shown to be impossible, when F. Lindemann proved that the ratio of the length of a circle to its diameter (denoted by π) is not the root of any algebraic equation with integer coefficients. Electronic computers have calculated π to over 1012 decimal places.
The area of a circle (circular disk) with radius r is πr2; the length (circumference) is 2πr. The area enclosed by a circle is greater than that bounded by any other curve of the same length. See also Analytic geometry; Conic section.
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underwriter’s way of designating potential purchasers and amounts of a securities issue during the Registration period, before selling is permitted. Registered representatives canvass prospective buyers and report any interest to the underwriters, who then circle the names on their list.
| Churning, Chunnel | |
| Circuit Breakers, Circus Swap |
Roget's Thesaurus:
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circle
noun
- A closed plane curve everywhere equidistant from a fixed point or something shaped like this: band, circuit, disk, gyre, ring, wheel. Archaic orb. See geometry.
- A course, process, or journey that ends where it began or repeats itself: circuit, cycle, orbit, round, tour, turn. See repetition.
- A group of people sharing an interest, activity, or achievement: crowd, group, set. See group.
- A particular social group: clique, coterie, crowd, set. Informal bunch, gang. See group.
- A sphere of activity, experience, study, or interest: area, arena, bailiwick, department, domain, field, orbit, province, realm, scene, subject, terrain, territory, world. Slang bag. See territory.
verb
- To shut in on all sides: begird, beset, compass, encircle, encompass, environ, gird, girdle, hedge, hem, ring, surround. See open/close.
- To move or cause to move in circles or around an axis: circumvolve, gyrate, orbit, revolve, rotate, turn, wheel. See move/halt, repetition.
Oxford Grove Music Encyclopedia:
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Circles
Work by Berio for female voice, harp and two percussionists (1960), settings of poems by E.E. Cummings; the singer moves to different positions on the platform.
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circling
Symbolically, a circle can stand for perfection, wholeness, or a boundary (protective or confining); circling round something can be a way of honouring or blessing it, or, conversely, of receiving blessing or power from it. Circling can also summon a supernatural being—it is one of the commonest English local traditions that if you run round a specified mound, tree, cross, grave, church, or stone at a specified time and/or a specified number of times without stopping, you will raise a ghost, or the Devil; the condition is less easy than it seems, since running round a small object causes giddiness, and round a large one is exhausting. The circle as boundary is exemplified by the common instruction in manuals of magic to draw a circle round oneself as protection against spirits summoned, or to conjure the spirit into a circle which will confine it; more prosaically, it appears also in the Devonshire belief that a snake cannot escape a circle drawn round it with an ash stick (Bray, 1838: 95).
See also LEFTWARD and RIGHTWARD MOVEMENT.
Columbia Encyclopedia:
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circles
circle, closed plane curve consisting of all points at a given distance from some fixed point, called the center. A circle is a conic section cut by a plane perpendicular to the axis of the cone. The term circle is also used to refer to the region enclosed by the curve, more properly called a circular region. The radius of a circle is any line segment connecting the center and a point on the curve; the term is also used for the length r of this segment, i.e., the common distance of all points on the curve from the center. Similarly, the circumference of a circle is either the curve itself or its length of arc. A line segment whose two ends lie on the circumference is a chord; a chord through the center is the diameter. A secant is a line of indefinite length intersecting the circle at two points, the segment of it within the circle being a chord. A tangent to a circle is a straight line touching the circle at only one point, the point of contact, or tangency, and is always perpendicular to the radius drawn to this point. A circle is inscribed in a polygon if each side of the polygon is tangent to the circle; a circle is circumscribed about a polygon if all the vertices of the polygon lie on the circumference. The length of the circumference C of a circle is equal to π (see pi) times twice the radius distance r, or C=2πr. The area A bounded by a circle is given by A=πr2. Greek geometry left many unsolved problems about circles, including the problem of squaring the circle, i.e., constructing a square with an area equal to that of a given circle, using only a straight edge and compass; it was finally proved impossible in the late 19th cent. (see geometric problems of antiquity). In modern mathematics the circle is the basis for such theories as inversive geometry and certain non-Euclidean geometries. The circle figures significantly in many cultures. In religion and art it frequently symbolizes heaven, eternity, or the universe.
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The circle is the most common space created for the working of magic and witchcraft. It stands in sharp contrast to the rectangular space that the average Christian church defines. The circle is easily drawn on the ground and just as easily erased. The circle has been a popular form for worship since ancient times as demonstrated by numerous stone monuments found around the world.
In modern magical and Wiccan practice, the circle is seen as both a protective barrier and a container of energy. It is the visible manifestation of a sphere that completely surrounds the worker of magic. Where the invisible sphere intersects the ground or floor, a circle is defined. While occasionally a more permanent circle is drawn and remains for regular workings, the circle is usually created only at the beginning of a magical ritual and dissolved at its close.
Modern magical rituals begin with the imaginal setting of a sphere of energy around the individual or group performing the ritual. Commonly, there are specific words that are spoken to create the sphere or circle. Most Wiccans believe in the existence of an array of spirit beings, from deities to elemental spirits. Most rituals are designed to invoke one or more of these deities and the intrusion of unwanted entities would disturb the focus of the ritual. In such settings, the circle is seen as a barrier that protects the ritual and keeps entities attracted by the power raised by the ritual from disturbing its fruitful conclusion. The ritual is closed with a banishing act dispersing any attending entities.
Modern rituals are also seen as acts that raise, focus, and direct energy to a specific purpose such as the healing of someone or the gaining of some particular favor. In such thinking, the sphere or circle is seen as a container that holds the energy so raised until the ritual's climax, when it is sent forth to do its work.
In modern Neo-Paganism, where worship predominates over magic, the idea of creating the circles as the creation of sacred space, apart from the mundane world, predominates. Sacred space is, or becomes, space in which the veil dividing the common everyday world from the realm of spirits is thin and communication is possible. While some sacred space is defined by the environment, a particularly beautiful or striking spot, it can be created anywhere. In the pantheistic Pagan world, all space is ultimately seen as sacred. Sacred space is often entered only after participants have cleansed themselves and donned special dress, commonly a ritual robe, or as in the case of some Wiccan groups, in the nude.
Sources:
Adler, Margot. Drawing Down the Moon. Boston: Beacon Press, 1979.
Crowley, Vivianne. Principles of Paganism. London: Thor-sons, 1996.
McGraw-Hill Slang Dictionary:
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circling (the drain)
tv. & in. to be in the final process of dying; to be in extremis. (Jocular but crude hospital jargon.) Get Mrs. Smith's son on the phone. She's circling the drain.
Word Tutor:
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circle
IN BRIEF: A closed curved line forming a perfectly round, flat figure.
The art project started with the drawing of a circle in the center of the paper.
Tutor's tip: The mice formed a "circle" (a curved line, every point of which is equally distant from the center) around the "cereal" (any grain used for food).
LearnThatWord.com is a free vocabulary and spelling program where you only pay for results!
Sign Language Videos:
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circle
sign description: The index finger makes a circular motion in the air.
The Dream Encyclopedia:
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Circle
A circle encompasses many meanings in numerous areas: the wholeness of numbers in mathematics, the spiritual oneness depicted by the circle and the mandala, protection from evil by the ritual drawing of a circle, bringing attention to something by circling it. It may also express frustrations, as when one doodles in circles or goes around in circles. Socially, it may represent being "in" the right circle of friends. The love relationship is sometimes symbolized by the wearing of a ring, around the finger, the neck, or in the nose. In Jungian psychology the circle is a symbol of the self archetype. (See also Zero)
Saunders Veterinary Dictionary:
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circling
Persistent walking in circles; it may be caused by deviation of the head because of a bend in the neck, or be due to rotation of the head, e.g. caused by listeriosis or brain abscess. A sign of unilateral vestibular disease or cerebral lesion (toward the side of the lesion; adverse syndrome).
- c. disease — see listeriosis.
Random House Word Menu:
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categories related to 'circling'
- Aviation and Aerodynamics - circling: flight pattern adopted by pilot while awaiting landing clearance or to position aircraft for landing
See crossword solutions for the clue Wikipedia on Answers.com:
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Circle
This article is about the shape and mathematical concept. For other uses, see Circle (disambiguation).
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This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (November 2010) |
| Circle | |
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 Circle illustration showing a radius, a diameter, the centre and circumference |
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A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are equidistant from a given point, the centre. The distance between any of the points and the centre is called the radius.
Circles are simple closed curves which divide the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is the former and the latter is called a disk.
A circle can be defined as the curve traced out by a point that moves so that its distance from a given point is constant.
A circle may also be defined as a special ellipse in which the two foci are coincident and the eccentricity is 0. Circles are conic sections attained when a right circular cone is intersected by a plane perpendicular to the axis of the cone.
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Terminology
A circle's diameter is the length of a line segment whose endpoints lie on the circle and which passes through the centre. This is the largest distance between any two points on the circle. The diameter of a circle is twice the radius, or distance from the centre to the circle's boundary. The terms "diameter" and "radius" also refer to the line segments which fit these descriptions. The circumference is the distance around the outside of a circle.
A chord is a line segment whose endpoints lie on the circle. A diameter is the longest chord in a circle. A tangent to a circle is a straight line that touches the circle at a single point, while a secant is an extended chord: a straight line cutting the circle at two points.
An arc of a circle is any connected part of the circle's circumference. A sector is a region bounded by two radii and an arc lying between the radii, and a segment is a region bounded by a chord and an arc lying between the chord's endpoints.
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History
The word "circle" derives from the Greek, kirkos "a circle," from the base ker- which means to turn or bend. The origins of the words "circus" and "circuit" are closely related.
The circle has been known since before the beginning of recorded history. Natural circles would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand. The circle is the basis for the wheel, which, with related inventions such as gears, makes much of modern civilisation possible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy, and calculus.
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.[citation needed]
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Tughrul Tower from inside
Circles on an old Arabic astronomical drawing
Some highlights in the history of the circle are:
- 1700 BC – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256 / 81 (3.16049...) as an approximate value of π.[1]
- 300 BC – Book 3 of Euclid's Elements deals with the properties of circles.
- 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.
- 1880 – Lindemann proves that π is transcendental, effectively settling the millennia-old problem of squaring the circle.[2]
Analytic results
Length of circumference
Further information: Pi
The ratio of a circle's circumference to its diameter is π (pi), an irrational constant approximately equal to 3.141592654. Thus the length of the circumference C is related to the radius r and diameter d by:
Area enclosed
Main article: Area of a disk
As proved by Archimedes, 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,[3] which comes to π multiplied by the radius squared:
Equivalently, denoting diameter by d,
that is, approximately 79 percent 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.
Equations
Cartesian coordinates
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 of the circle follows from the Pythagorean theorem applied to any point on the circle: as shown in the diagram to the right, 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 x-axis. An alternative parametrisation of the circle is:
In this parametrisation, the ratio of t to r can be interpreted geometrically as the stereographic projection of the circle onto the line passing through the centre parallel to the x-axis.
In homogeneous coordinates each conic section with equation of a circle is of 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, 
is the polar coordinate of a generic point on the circle, and 
is the polar coordinate 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 at the origin, i.e. r0 = 0, this reduces to simply r = a. When r0 = a, or when the origin lies on the circle, the equation becomes
In the general case, the equation can be solved for r, giving
the solution with a minus sign in front of the square root giving the same curve.
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 
.
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
Main article: Tangent lines to circles
The tangent line through a point P on the circle is perpendicular to the diameter passing through P. If P = (x1, y1) 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 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
- 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.
- A circle's 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 which 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:
- A perpendicular line from the centre of a circle bisects the chord.
- The line segment (circular 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 supplemental.
- 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.
- 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 a2 + b2 + c2 + d2 equals the square of the diameter.[4]
- 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 chords intersecting at the same point, and is given by 8r 2 – 4p 2 (where r is the circle's radius and p is the distance from the center point to the point of intersection).[5]
- 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.[6]:p.71
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 which 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 it is 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.
Tangent
- The line perpendicular drawn to a radius through the end point of the radius 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).
Theorems
See also: Power of a point
- The chord theorem states that if two chords, CD and EB, intersect at A, then CA × DA = EA × BA.
- If a tangent from an external point D meets the circle at C and a secant from the external point D meets the circle at G and E respectively, then DC2 = DG × DE. (Tangent-secant theorem.)
- If two secants, DG and DE, also cut the circle at H and F respectively, then DH × DG = DF × DE. (Corollary of the tangent-secant theorem.)
- The angle between a tangent and chord is equal to one half the subtended angle on the opposite side of the chord (Tangent Chord Angle).
- If the angle subtended by the chord at the centre is 90 degrees 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 (DE and BC). This is the secant-secant theorem.
Inscribed angles
See also: 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 degrees).
Circle of Apollonius
Main article: Apollonian circles
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.[7][8] (The set of points where the distances are equal is the perpendicular bisector of A and B, 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, i.e., 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[9]:p.15 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:
![|[A,B;C,P]| = 1.\](http://wpcontent.answcdn.com/wikipedia/en/math/9/e/0/9e035e9d62f94ddf78be8760a6482aab.png)
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
See also: Generalised circle
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.
Circles inscribed in or circumscribed 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.[10]
About every triangle a unique circle, called the circumcircle, can be circumscribed such that it goes through each of the triangle's three vertices.[11]
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.[12]
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.
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.
Circle as limiting case of other figures
The circle can be viewed as a limiting case of each of various other figures:
- 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.
See also
References
- ^ Chronology for 30000 BC to 500 BC. History.mcs.st-andrews.ac.uk. Retrieved on 2012-05-03.
- ^ Squaring the circle. History.mcs.st-andrews.ac.uk. Retrieved on 2012-05-03.
- ^ Measurement of a Circle by Archimedes
- ^ 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. London, New York: Macmillan and Co.. p. 30. http://dlxs2.library.cornell.edu/cgi/t/text/text-idx?c=math;idno=01680002.
- ^ Ogilvy, C. Stanley, Excursions in Geometry, Dover, 1969, 14–17.
- ^ Altshiller-Court, Nathan, College Geometry, Dover, 2007 (orig. 1952).
- ^ Incircle – from Wolfram MathWorld. Mathworld.wolfram.com (2012-04-26). Retrieved on 2012-05-03.
- ^ Circumcircle – from Wolfram MathWorld. Mathworld.wolfram.com (2012-04-26). Retrieved on 2012-05-03.
- ^ Tangential Polygon – from Wolfram MathWorld. Mathworld.wolfram.com (2012-04-26). Retrieved on 2012-05-03.
Further reading
- Pedoe, Dan (1988). Geometry: a comprehensive course. Dover.
- "Circle" in The MacTutor History of Mathematics archive
External links
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Wikimedia Commons has media related to: Circle geometry |
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Wikiquote has a collection of quotations related to: Circles |
- Circle (PlanetMath.org website)
- Weisstein, Eric W., "Circle" from 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 at cut-the-knot
- Area of a Circle Calculate the basic properties of a circle.
- MathAce's Circle article – has a good in-depth explanation of unit circles and transforming circular equations.
This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)
Translations:
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Circle
Dansk (Danish)
n. - cirkel, rundkreds, omkreds, kreds, rotunde
v. tr. - kredse om, tegne cirkel om, omgive
v. intr. - kredse
Nederlands (Dutch)
kring, cirkel(-), rotonde, balkon (theater), slagcirkel, cyclus, omcirkelen, omlijnen
Français (French)
n. - cercle, rond, groupe, (Théât) balcon
v. tr. - tourner autour de, graviter autour de, faire le tour de, tourner autour de (qn, animal), encercler
v. intr. - tourner en rond autour de, décrire des cercles au-dessus de
Deutsch (German)
n. - Kreis, Rang, Kreisel
v. - einkreisen, umkreisen
Ελληνική (Greek)
n. - κύκλος, ανακύκληση, κοινωνικός κύκλος, κοινωνικός περίγυρος, εξώστης θεάτρου
v. - ζώνω, κυκλώνω, περικυκλώνω, περιτριγυρίζω, περιστρέφομαι, στριφογυρίζω
idioms:
- high circles υψηλοί κύκλοι, της καλής κοινωνίας
Italiano (Italian)
circondare, cerchia, circolo, cerchio, balconata, galleria, rotonda, circolare
idioms:
- come full circle/ turn full circle ritornare al punto di partenza
- go round in a circle girare in tondo
Português (Portuguese)
n. - círculo (m), circunferência (f), anel (m), coroa (f), giro (m), ciclo (m), rodeio (m), órbita (f), âmbito (m), ciclo (m) social, divisão (f) territorial
v. - circundar, girar em torno, mover-se em círculos
idioms:
- come/turn full circle voltar ao início
- dress circle balcão (m) nobre
- go round in a circle não progredir
- high circles altas rodas (f pl)
- inner circle grupo (m) fechado
Русский (Russian)
описывать круги, обводить кружком, круг, цикл, бельэтаж
idioms:
- come/turn full circle описать полный круг
- dress circle бельэтаж
- go round in a circle вертеться как белка в колесе
- high circles высшие круги
- inner circle наиболее близкие друзья, приближенные
Español (Spanish)
n. - círculo, corro, rueda, anillo, piso principal del teatro, rotonda, circular
v. tr. - cercar, rodear
v. intr. - poner un círculo alrededor de, rodearse
Svenska (Swedish)
n. - cirkel, omkrets, kretsgång, full serie, krets, rad (teat.), rund stensättning (arkeol.)
v. - omge, gå/fara omkring, ringa in, kretsa
中文（简体）(Chinese (Simplified))
圆周, 循环, 社交圈, 包围, 环绕, 盘旋, 旋转, 流传, 环行
中文（繁體）(Chinese (Traditional))
n. - 圓周, 循環, 社交圈
v. tr. - 包圍, 環繞
v. intr. - 盤旋, 旋轉, 流傳, 環行
한국어 (Korean)
n. - 원, 주기, 집단
v. tr. - 선회하다, 에워싸다, 우회하다
v. intr. - 돌다
日本語 (Japanese)
n. - 円, 仲間, 範囲, 周期, 桟敷, 圏, 輪, 集団, 全系統, 悪循環
v. - 回る, 一周する, 丸で囲む, 回転する
العربيه (Arabic)
(الاسم) دائرة, حلقه (فعل) حام, طاف, أحاط, دور
עברית (Hebrew)
n. - מחזור, חוג, טבעת, מעגל, עיגול, גוש מושבים בתיאטרון
v. tr. - הקיף
v. intr. - הסתובב, חג
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