Circles are simple closed curves which divide the plane into two regions, one inside and one outside. In everyday use, the term "circle" are used interchangeably to refer to either boundary of the figure (known as the circumference) or to whole numbers, including its interior. But in the narrow technical use, which means "circle" to the edge, while the interior of the circle is called a disc. The circumference of a circle is the circumference of a circle (especially when referring to its length).
A circle is a
special ellipse in which the two foci coincide. Circles are conic sections attained when a right circular cone is intersected with a plane perpendicular to the cone.
Additional terminology:
The diameter of a circle is the length of a segment whose end points lie on the circle and passes through the center of the circle. This is the largest distance between two points on the circle. The diameter of a circle is twice the radius.
The term "radius" can also refer to a segment from the center of a circle with a circumference, and also the term "diameter" may refer to a segment between two points on the edge, which passes through the center. In this way, the midpoint of a diameter in the middle, and so it is composed of two radii.
A chord of a circle is a line whose two endpoints located on the circle. Diameter passing through its center, is the greatest chord in a circle. A tangent to a circle is a straight line touching the circle at a time. A secant is an extended chord: a straight line cutting the circle in two points.
A circle is all connected to a part of the circle's circumference. A sector is an area bounded by two radii and an arc lying between the radii and a segment is an area bounded by a chord and an arc lying between the chord's endpoints.
Features:
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 center forms a line of reflection symmetry and it has rotational symmetry around the center of 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 circumference and radius are proportional.
The area enclosed and the square root of its radius are proportional.
Constants of proportionality are 2π and π, respectively.
Circle centered at the origin with radius 1 is called the unit circle.
Conceived as a great circle of the unit sphere, it becomes Riemannian circle.
Through all three points not all on the same line, there is a unique circle. In Cartesian
coordinates, it is possible to give explicit formulas for the coordinates of its center and the radius in the form of coordinates of the three given points. See circumcircle.
Chord:
Chords are equidistant from the center of a circle if and only if they are of equal length.
The perpendicular bisector of a chord passes through the center of a circle, similar statements from the uniqueness of the perpendicular bisector:
A vertical line from the center of a circle bisects the chord.
The line segment (Circular segment) through the center runs a chord is perpendicular to the chord.
If a central angle and an angle inscribed in a circle where during the same chord and on
the same side of the chord, so 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, so they are equal.
If two angles are inscribed on the same chord and on opposite sides of the chord, so they complement each other.
For a cyclic square is the exterior angle is equal to the interior opposite angle.
An inscribed angle in which a diameter is a right angle.
The diameter is the longest chord of the circle.
Sagitta:
The Sagitta (also known as versin) is a segment drawn perpendicular to a chord, between the midpoint of this chord and the circumference of the circle.
Because of the length y of a chord, and the length x of Sagitta, the Pythagorean theorem can be used to calculate the radius of the unique circle that fits around the two lines:
Another proof of this result which relies only on two chord properties above are as follows. Given a chord of length y and with Sagitta of length x as Sagitta intersects the midpoint of the chord, and we know that it is part of a diameter of the circle. Since the diameter is twice the radius, the "missing" part of the diameter (2r - x) in length. Using the fact that a portion of one chord times the other side is equal to the same product taken along a chord intersects the first chord, we find that (2r - x) x = (y / 2) ². Solve for r, we find the desired result.
Tangent:
The line perpendicular to a radius of the end point of the radius is a tangent to the circle.
A line perpendicular to a tangent through the point of contact with a circle that goes through the center of the circle.
Two tangents can always be aware of a circle from any point outside the circle, and these tangents are equal length.
Sentences:
Chord sentence 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 circle C and a secant from the external point
D meets circle G and E, as 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 equals degree angle on the opposite side of the chord. (Tangent chord property.)
If the angle at which 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 measure the angle A is equal to half the difference between the measurements of the enclosed arcs (DE and BC).
This is the secant-secant theorem.
Inscribed angles:
An inscribed angle (examples are the blue and green dots in figure) is exactly half the corresponding central angle (red). Therefore, all inscribed angles that subtend the same arc (pink) are similar. Angles inscribed on the arc (brown) is a complement. In particular, subtends each enrolled angular diameter is a right angle (since the central angle is 180 degrees).
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