A tangent can be considered a limiting case of a secant whose ends are coincident. If a tangent from an external point A meets the circle at F and a secant from the external point A meets the circle at C and D respectively, then AF 2 = AC × AD (tangent–secant theorem). The centre of the circle is the fixed point from which all points on the boundary of the circle are equidistant.

About every triangle a unique circle, called the circumcircle, can be circumscribed such that it goes through each of the triangle's three vertices. [17]

Diameter of a circle

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. Every regular polygon and every triangle is a cyclic polygon. A polygon that is both cyclic and tangential is called a bicentric polygon.

The circle is the shape with the largest area for a given length of perimeter (see Isoperimetric inequality). 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. [18] Every regular polygon and every triangle is a tangential polygon. 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 that it is a 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 the required result. However, differences in worldview (beliefs and culture) had a great impact on artists’ perceptions. While some emphasised the circle's perimeter to demonstrate their democratic manifestation, others focused on its centre to symbolise the concept of cosmic unity. In mystical doctrines, the circle mainly symbolises the infinite and cyclical nature of existence, but in religious traditions it represents heavenly bodies and divine spirits.Note: Secant is not a term you are required to know at GCSE, however it is important to note the difference between a chord and a secant. Semicircle: one of the two possible arcs determined by the endpoints of a diameter, taking its midpoint as centre. In non-technical common usage it may mean the interior of the two-dimensional region bounded by a diameter and one of its arcs, that is technically called a half-disc. A half-disc is a special case of a segment, namely the largest one. A superellipse has an equation of the form | x a | n + | y b | n = 1 {\displaystyle \left|{\frac {x}{a}}\right| A line segment going from one point of the circumference to another but does not go through the centre.

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. Arc: any connected part of a circle. Specifying two end points of an arc and a centre allows for two arcs that together make up a full circle. If two secants, AE and AD, also cut the circle at B and C respectively, then AC × AD = AB × AE (corollary of the chord theorem). From the time of the earliest known civilisations – such as the Assyrians and ancient Egyptians, those in the Indus Valley and along the Yellow River in China, and the Western civilisations of ancient Greece and Rome during classical Antiquity – the circle has been used directly or indirectly in visual art to convey the artist's message and to express certain ideas. The simplest and most basic is the construction given the centre of the circle and a point on the circle. Place the fixed leg of the compass on the centre point, the movable leg on the point on the circle and rotate the compass.

Names of parts of a circle

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 perpendicular chords intersecting at the same point and is given by 8 r 2 − 4 p 2, where r is the circle radius, and p is the distance from the centre point to the point of intersection. [11]

If two angles are inscribed on the same chord and on the same side of the chord, then they are equal. Sector: a region bounded by two radii of equal length with a common centre and either of the two possible arcs, determined by this centre and the endpoints of the radii.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 ( D E ⌢ {\displaystyle {\overset {\frown }{DE}}} and B C ⌢ {\displaystyle {\overset {\frown }{BC}}} ). That is, 2 ∠ C A B = ∠ D O E − ∠ B O C {\displaystyle 2\angle {CAB}=\angle {DOE}-\angle {BOC}} , where O is the centre of the circle (secant–secant theorem). Among all the circles with a chord AB in common, the circle with minimal radius is the one with diameter AB. Radius: a line segment joining the centre of a circle with any single point on the circle itself; or the length of such a segment, which is half (the length of) a diameter. Usually, the radius is denoted r {\displaystyle r} and required to be a positive number. A circle with r = 0 {\displaystyle r=0} is a degenerate case consisting of a single point.