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# 6 Circles

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

Conversely, an equation of the form

x+y+2dx+2ey+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 PP; its equation is

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

The length or circumference of a circle of radius r is 2r, 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 1i,j4, 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:

±dd±dd±dd=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=2a cos(-).

A circle of radius a and with center at the point (r,)=(r,) has equation

r-2rr cos(-)+r-a=0.

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Up: Part I: Two-Dimensional Geometry
Previous: 5.3 Regular Polygons