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Encyclopedia > Circle
Circle illustration showing a radius, a diameter, the centre and the circumference.
Tycho crater, one of many examples of circles that arise in nature. NASA photo.

A chord of a circle is a line segment whose both endpoints lie on the circle. A diameter is a chord passing through the center. The length of a diameter is twice the radius. A diameter is the largest chord in a circle. A chord of a curve is a geometric line segment whose endpoints both lie on the curve. ... DIAMETER is a computer networking protocol for AAA (Authentication, Authorization and Accounting). ...

Circles are simple closed curves which divide the plane into an interior and an exterior. The circumference of a circle is the perimeter of the circle, and the interior of the circle is called a disk. An arc is any connected part of a circle. In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... The circumference is the distance around a closed curve. ... In geometry, a disk is the region in a plane contained inside of a circle. ... In Euclidean geometry, an arc is a closed segment of a differentiable curve in the two-dimensional plane; for example, a circular arc is a segment of a circle. ... Connected and disconnected subspaces of RÂ². The space A at top is connected; the shaded space B at bottom is not. ...

A circle is a special ellipse in which the two foci are coincident. Circles are conic sections attained when a right circular cone is intersected with a plane perpendicular to the axis of the cone. For other uses, see Ellipse (disambiguation). ... In geometry, the focus (pl. ... Wikibooks has more on the topic of Conic section Types of conic sections Table of conics, Cyclopaedia, 1728 In mathematics, a conic section (or just conic) is a curve that can be formed by intersecting a cone (more precisely, a right circular conical surface) with a plane. ... In geometry, a (general) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point â€” the apex or vertex â€” and any point of some fixed space curve â€” the directrix â€” that does not contain the apex. ...

• The circle is the shape with the largest area for a given length of perimeter. (See Isoperimetry)
• 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.
• The circle centered at the origin with radius 1 is called the unit 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 center of the circle and the radius in terms of the coordinates of the three given points. See circumcircle.

### Chord properties

• Chords equidistant from the center of a circle are equal (length).
• Equal (length) chords are equidistant from the center.
• The perpendicular bisector of a chord passes through the center of a circle; equivalent statements stemming from the uniqueness of the perpendicular bisector:
• A perpendicular line from the center of a circle bisects the chord.
• The line segment (Circular segment) through the center 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.
• The diameter is longest chord of the circle.

The geometric definition of a line segment In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. ... In geometry, a circular segment (also circle segment) is an area of a circle informally defined as an area which is cut off from the rest of the circle by a secant or a chord. ... In geometry, an inscribed angle is formed when two secant lines of a circle (or, in a degenerate case, when one secant line and one tangent line of that circle) intersect on the circle. ...

### Sagitta properties

• The sagitta is a line segment drawn perpendicular to a chord, between the midpoint of that chord and the circumference 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:
$r=frac{y^2}{8x}+ frac{x}{2}.$

Another proof of this result which relies only on 2 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 (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: In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ...

$r=frac{y^2}{8x}+ frac{x}{2}.$

as required.

### Tangent properties

• The line drawn perpendicular to the end point of a radius is a tangent to the circle.
• A line drawn perpendicular to a tangent at the point of contact with a circle passes through the centre of the circle.
• Tangents drawn from a point outside the circle are equal in length.
• Two tangents can always be drawn from a point outside of the circle.

### Theorems

Secant-secant theorem
• The chord theorem states that if two chords, CD and EB, intersect at A, then CA×DA = EA×BA. (Chord theorem)
• 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 the subtended angle on the opposite side of the chord. (Tangent chord property)
• If the angle subtended by the chord at the center is 90 degrees then l = √2 × r, 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.

Image File history File links Secant-Secant_Theorem. ... Image File history File links Secant-Secant_Theorem. ... The power of a point A (circle power,power of a circle) with respect to a circle with center 0 and radius r is defined as Therefore points inside the circle have negative power, points outside have positive power, and points on the circle have power zero. ... Secant is a term in mathematics. ... This article describes the unit of angle. ...

### Inscribed angles

Inscribed angle theorem

An inscribed angle ψ is exactly half of the corresponding central angle θ (see Figure). Hence, all inscribed angles that subtend the same arc have the same value (cf. the blue and green angles ψ in the Figure). Angles inscribed on the arc are supplementary. In particular, every inscribed angle that subtends a diameter is a right angle. Image File history File links Inscribed_angle_theorem. ... Image File history File links Inscribed_angle_theorem. ... In geometry, an inscribed angle is formed when two secant lines of a circle (or, in a degenerate case, when one secant line and one tangent line of that circle) intersect on the circle. ... Angle AOB forms a central angle of circle O A central angle is an angle whose vertex is the center of a circle, and whose sides pass through a pair of points on the circle, thereby subtending an arc between those two points whose angle is (by definition) equal to... DIAMETER is a computer networking protocol for AAA (Authentication, Authorization and Accounting). ... This article is about angles in geometry. ...

## Apollonius circle

$frac{d_1}{d_2}=textrm{constant}$ Apollonius' definition of a circle

Apollonius of Perga showed that a circle may also be defined as the set of points in plane having a constant ratio of distances to two fixed foci, A and B. Image File history File links Apollonius_circle_definition_labels. ... Image File history File links Apollonius_circle_definition_labels. ... Apollonius of Perga [Pergaeus] (ca. ...

The proof is as follows. A line segment PC bisects the interior angle APB, since the segments are similar:

$frac{AP}{BP} = frac{AC}{BC}.$

Analogously, a line segment PD bisects the corresponding exterior angle. Since the interior and exterior angles sum to $180^{circ}$, the angle CPD is exactly $90^{circ}$, i.e., a right angle. The set of points P that form a right angle with a given line segment CD form a circle, of which CD is the diameter.

### Cross-ratios

Early science, particularly geometry and astronomy/astrology, was connected to the divine for most medieval scholars. Notice, even, the circular shape of the halo. The compass in this 13th century manuscript is a symbol of God's act of Creation, as many believed that there was something intrinsically "divine" or "perfect" that could be found in circles

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 Apollonius circle for these three points is the collection of points P for which the absolute value of the cross-ratio is equal to one: Image File history File links God_the_Geometer. ... Image File history File links God_the_Geometer. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... For other uses, see Geometry (disambiguation). ... This article needs additional references or sources for verification. ... The history of science in the Middle Ages refers to the discoveries in the field of natural philosophy throughout the Middle Ages - the middle period in a traditional schematic division of European history. ... Look up halo, HALO in Wiktionary, the free dictionary. ... a compass In drafting, a compass (or pair of compasses) is an instrument]] used by mathematicians and craftsmen in for drawing or inscribing a circle or arc. ... (12th century - 13th century - 14th century - other centuries) As a means of recording the passage of time, the 13th century was that century which lasted from 1201 to 1300. ... THIS IS A FACT Creation is a doctrinal position in many religions and philosophical belief systems which maintains that a single God, or a group of or deities is responsible for creating the universe. ... In mathematics, the cross-ratio cr( w, x, y, z ) of an ordered quadruple of complex numbers (which may be real numbers) is Cross-ratios are preserved by linear fractional transformations, i. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...

| [A,B;C,P] | = 1.

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

### Generalized circles

If C is the midpoint of the segment AB, then the collection of points P satisfying the Apollonius condition A circle Î“ is the set of points p that lie at radius r from a center point Î³. Using the complex plane, we can treat Î³ as a complex number and circle Î“ as a set of complex numbers. ... In mathematics, a line segment is a part of a line that is bounded by two end points. ...

$frac{|AP|}{|BP|} = frac{|AC|}{|BC|}$   (1)

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 (1) is called a generalized circle. It may either be a true circle or a line.

## References

• ^ Pedoe, Dan (1988). Geometry: a comprehensive course. Dover.

Wikimedia Commons has media related to:

This list of circle topics is not intended for metaphorical circles, but rather for topics related to the geometric shape. ...

Results from FactBites:

 Circle Sanctuary Website (298 words) Founded in 1974 by Selena Fox, Circle Sanctuary is a non-profit organization dedicated to networking, research, spiritual healing, community celebrations, and education. The Circle sponsors gatherings and encourages Nature preservation with Nature meditations and workshops celebrated at diverse locations including the Circle Sanctuary Nature Preserve, a 200 acre site in the forested hills of southwestern Wisconsin, USA. Circle Sanctuary maintains the Circle Network, an information exchange and contact service which links together individuals, groups, networks, centers, and other groupings of Pagans/Nature Spirituality practitioners throughout the world.
 Area of a Circle (332 words) (pronounced Pi) to represent the ratio of the circumference of a circle to the diameter. The diameter of a circle is 8 centimeters. The radius of a circle is 9 centimeters.
More results at FactBites »

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