circle - Analytic results, Properties, Inscribed angles, An alternative definition of a circle, Numbers and the circle

The locus of a point that is a constant distance from a fixed point. The constant distance is called the radius, and the fixed point the centre. From this definition are obtained all the properties of the circle. Thus if A and B are two points on a circle centre O, triangle OAB is isosceles, and so the perpendicular bisector of any chord of a circle passes through the centre of the circle. By the use of two isosceles triangles, it can be proved that the angle subtended by an arc of a circle at the centre of a circle is twice the angle subtended by that arc at a point on the circumference of the circle. The area of a circle radius r is ?r²; the circumference is 2?r. Archimedes c.240 BC showed that 223/71 < ? < 22/7.

This article is about the shape and mathematical concept of circle. For other uses, 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. Usually, however, the circumference means the length of the circle, and the interior of the circle is called a disk.

Analytic results

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

.

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

x = r2.

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

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:

.

In the complex plane, a circle with a centre at c and radius r has the equation | 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.

In other words:

Length of a circle's circumference is: Area of a circle is: Diameter of a circle is:

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. 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.

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.

In Chinese mathematics the area of a circle is usually expressed as:

A=c/2•d/2

Where A is the area, c is the circumference, and d is the diameter.

Properties

The circle is the shape with the highest area for a given length of perimeter. 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 centre for every angle.

Chord properties

Chords equidistant from the centre of a circle are equal (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. 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.

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. (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 . (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. If two secants are inscribed in the circle like so

then the measurement of angle A is equal to one half the difference of the measurements of the enclosed arcs (DE and BC).

Inscribed angles

An inscribed angle ψ is exactly half of the corresponding central angle θ (see Figure).

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. A line segment PC bisects the interior angle APB, since the segments are similar

Analogously, a line segment PD bisects the corresponding exterior angle. Since the interior and exterior angles sum to , the angle CPD is exactly , 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.

Numbers and the circle

The division of the circle into 360 degrees dates back to ancient India, as found in the Rig Veda:

Twelve spokes, one wheel, navels three.

The enneagram expresses the circle as equal to 9 integers. The structure of the enneagram is based partly on the primary triangle of the circle at 0/360 degrees, 120 degrees and 240 degrees of the circle.