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The set of points whose distance to a fixed point (the **center**) is
a fixed positive number (the **radius**) is a **circle**.
A circle of radius *r* and center (*x*,*y*) has equation

(*x*-*x*)+(*y*-*y*)=*r*,

or

*x*+*y*-2*xx*-2*yy*+*x*+*y*-*r*=0.

Conversely, an equation of the form

*x*+*y*+2*dx*+2*ey*+*f*=0

defines a circle if *d*+*e*>*f*; the center is (-*d*, -*e*)
and the radius is .

Three points not on the same line determine a unique circle.
If the points have coordinates (*x*,*y*),
(*x*,*y*) and (*x*,*y*), the equation
of the circle is

A **chord** of a circle is a line segment between two points
(Figure 1). A **diameter** is a chord that goes through
the center, or the length of such a chord (therefore the diameter is
twice the radius). Given two points *P*=(*x*,*y*) and
*P*=(*x*,*y*), there is a unique circle whose diameter is
*P**P*;
its equation is

(*x*-*x*)(*x*-*x*)+(*y*-*y*)(*y*-*y*)=0.

The **length** or **circumference** of a circle of radius *r* is
2*r*, and the **area** is *r*. The length of the **arc
of circle** subtended by an angle , shown as *s* in
Figure 1, is *r*. Other relations between the
radius, the arc length, the chord, and the areas of the corresponding
**sector** and **segment** are, in the notation of
(Figure 1):

**Figure 1:** The arc of circle subtendend by the angle is *s*;
the chord is *c*; the sector is the whole slice of the pie; the
segment is the cap bounded by the arc and the chord (that is, the
slice minus the triangle).

Other properties of circles:

- If a line intersects a circle of center
*O*at points*A*and*B*, the segments*OA*and*OB*make equal angles with the line. In particular, a tangent line is perpendicular to the radius that goes through the point of tangency. - Given a fixed circle and a fixed point
*P*in the plane, and a line through*P*that intersects the circle at*A*and*B*(with*A=B*for a tangent). Then the product of the distances*PA*×*PB*is the same for all such lines. It is called the**power**of*P*with respect to the circle. - If the central angle
*AOB*equals , the angle*ACB*, where*C*is any point on the circle, equals ½ or 180°-½ (Figure 2, left). Conversely, given a segment*AB*, the set of points that ``sees''*AB*under a fixed angle is an arc of a circle (Figure 2, right). In particular, the set of points that see*AB*under a right angle is a circle with diameter*AB*.**Figure 2:**Left: The angle*ACB*equals ½ for any*C*in the long arc*AB*; and*ADB*equals 180°-½ for any*D*in the short arc*AB*. Right: The locus of points from which the segment*AB*subtends a fixed angle is an arc of circle.

- Let
*P*,*P*,*P*,*P*be points in the plane, and let , for 1*i*,*j*4, be the distance between and . A necessary and sufficient condition for the points to all lie on the same circle (or line) is that one of the following equalities be satisfied:±

*d**d*±*d**d*±*d**d*=0.This is equivalent to Ptolemy's formula for cyclic quadrilaterals .

- In
**oblique coordinates**with angle , a circle of center (*x*,*y*) and radius*r*has equation(

*x*-*x*)+ (*y*-*y*)+ 2(*x*-*x*)(*y*-*y*) cos =*r*. - In
**polar coordinates**, a circle centered at the pole and having radius*a*has equation*r*=*a*.A circle of radius

*a*, passing through the pole, and with center at the point (*r*,)=(*a*,) has equation*r*=2*a*cos(-).A circle of radius

*a*and with center at the point (*r*,)=(*r*,) has equation*r*-2*r**r*cos(-)+*r*-*a*=0.

*Silvio Levy
Wed Oct 4 16:41:25 PDT 1995*

This document is excerpted from the 30th Edition of the *CRC Standard Mathematical Tables and Formulas* (CRC Press). Unauthorized duplication is forbidden.