# A region of a circle contained by a line and a chord

In plane geometry, a chord is the line segment joining two points on a curve. The term is often used to describe a line segment whose ends lie on a circle.

The term is also used in graph theory, where a cycle chord of a graph cycle is an edge not in whose endpoints lie in .

In the above figure, is the radius of the circle, is the chord length, is called the apothem, and the sagitta.

The shaded region in the left figure is called a circular sector, and the shaded region in the right figure is called a circular segment.

There are a number of interesting theorems about chords of circles. All angles inscribed in a circle and subtended by the same chord are equal. The converse is also true: The locus of all points from which a given segment subtends equal angles is a circle.

In the left figure above,

(1)

(Jurgensen 1963, p.345). In the right figure above,

(2)

which is a statement of the fact that the circle power is independent of the choice of the line (Coxeter 1969, p.81; Jurgensen 1963, p.346).

Given any closed convex curve, it is possible to find a point through which three chords, inclined to one another at angles of , pass such that is the midpoint of all three (Wells 1991).

Let a circle of radius have a chord at distance . The area enclosed by the chord, shown as the shaded region in the above figure, is then

(3)

But

(4)

so

(5)

and

(6)
(7)

Checking the limits, when , and when ,

(8)

the expected area of the semicircle.