Circle

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This article is about the shape and mathematical concept of circle; for other meanings, see circle (disambiguation). In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, the centre. The points can only be those that are part of a conic section; within the set of a plane normal to the axis of a right cone. Circles are simple closed curves, dividing the plane into an interior and exterior. Sometimes the word circle is used to mean the interior, with the circle itself called the circumference(C). Usually, however, the circumference means the length of the circle, and the interior of the circle is called a disk. An arc is any continuous portion of a circle. Circles are named by their centre, i.e. Circle O or ʘE.

Mathematical definitions

In an x-y coordinate system, the circle with center (a, b) and radius r is the set of all points (x, y) such that

\left( x - a \right)^2 + \left( y - b \right)^2=r^2.

If the circle is centred at the origin (0, 0), then this formula can be simplified to

x^2 + y^2 = r^2.

The circle centred at the origin with radius 1 is called the unit circle,

Expressed in parametric equations, (x, y) can be written using the trigonometric functions sine and cosine as

x = a + r cos(t)

y = b + r sin(t).

The slope of a circle at a point (x, y) can be expressed with the following formula, assuming the centre is at the origin and (x, y) is on the circle:

y' = - \frac{x}{y}.

In the complex plane, a circle with a centre at c and radius r has the equation $|z-c|^2 = r^2$ . Since $|z-c|^2 = z\overline{z}-\overline{c}z-c\overline{z}+c\overline{c}$ , the slightly generalized equation $pz\overline{z} + gz + \overline{gz} = q$ for real p, q and complex g is sometimes called a generalized circle. It is important to note that not all generalized circles are actually circles.

All circles are similar; as a consequence, a circle's circumference and radius are proportional, as are its area and the square of its radius. The constants of proportionality are 2π and π, respectively.

In other words:

Length of a circle's circumference = $2\pi \times r.$
Area of a circle = $\pi \times r^2.$

The formula for the area of a circle can be derived from the formula for the circumference and the formula for the area of a triangle, as follows. Imagine a regular hexagon (six-sided figure) divided into equal triangles, with their apices at the centre of the hexagon. The area of the hexagon may be found by the formula for triangle area by adding up the lengths of all the triangle bases (on the exterior of the hexagon), multiplying by the height of the triangles (distance from the middle of the base to the center) and dividing by two. This is an approximation of the area of a circle. Then imagine the same exercise with an octagon (eight-sided figure), and the approximation is a little closer to the area of a circle. As a regular polygon with more and more sides is divided into triangles and the area calculated from this, the area becomes closer and closer to the area of a circle. In the limit, the sum of the bases approaches the circumference 2πr, and the triangles' height approaches the radius r. Multiplying the circumference and radius and dividing by 2, we get the area, π r².

The formula for the area of circle can also be derived by using an infinitesimal area element $dA$ and integrating it over the whole circle.

Properties

The circle is the shape with the highest area for a given length of perimiter.
The circle is a highly symetric shape, every line through the center forms a line of Reflectional symmetry and it has rotational symmetry around the centre for every angle. Its symmetry group is orthogonal group O(2,R). The group of rotations alone is the circle group T.

Chord properties

Chords equidistant from the centre of a circle are equal.
Equal chords are equidistant from the centre.
A line from the centre, perpendicular to a chord, bisects the chord.
The line segment through the center bisecting a chord is perpendicular to the chord.
The perpendicular bisector of a chord passes through the centre of a circle.

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 center 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.
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.
An inscribed angle subtended by a semicircle is a right angle.
For a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle.

Theorems

The chord theorem states that if two chords, CD and EF, intersect at G, then $CG \times DG = EG \times FG$ . (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 $DC^2 = DG \times DE$ . (tangent-secant theorem)
If two secants, DG and DE, also cut the circle at H and F respectively, then $DH \times DG = DF \times DE$ . (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 centre is 90 degrees then l = √(2) × r, where l is the length of the chord and r is the radius of the circle. then the measurement of angle A is equal to (measurement of arc DE - Measurement of arc BC)/2

Inscribed angles

An inscribed angle $\psi$ is exactly half of the corresponding central angle $\theta$ (see Figure). Hence, all inscribed angles that subtend the same arc have the same value (cf. the blue and green angles $\psi$ in the Figure). In particular, every inscribed angle that subtends a diameter is a right angle.

An alternative definition of a circle

Apollonius of Perga showed that a circle may also be defined as the set of points having a constant ratio of distances to two foci, A and B.

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

\frac{AP}{BP} = \frac{AC}{CB}

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.

External links

Clifford's Circle Chain Theorems. This is a step by step presentation of the first theorem. Clifford discovered, in the ordinary Euclidean plane, a "sequence or chain of theorems" of increasing complexity, each building on the last in a natural progression. by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
Munching on Circles at cut-the-knot

circles have ever ladting of simatry Geometric shapes | Curves | Conic sections | Pi

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