6 circles

8/16/2019 6 Circles

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 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 andOB make equalangles 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 circleat  A and B  (with A=B for a tangent). Then the product of the distances  PA× PB is the same for allsuch 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 afixed angle is an arc of a circle (Figure 2, right). In particular, the set of points that see  AB under aright angle is a circle with diameter  AB.

 Figure 2: Left: The angle ACB equals ½ for anyC  in the long arc  AB; and  ADB equals 180°-½