Chapter 4: Geometry

FIGURE 4.18
The arc of a circle subtended by the angle is ×; the chord is ; 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).
×
Ê
FIGURE 4.19
Left: the angle equals ½ for any in the long arc ; equals ½¼Æ
½ for
¾ ¾
any in the short arc . Right: the locus of points, from which the segment subtends
a xed angle , is an arc of the circle.
points to lie on the same circle (or line) is that one of the following equalities
be satis ed:
¦ ½¾ ¿ ¦ ½¿ ¾ ¦ ½ ¾¿ ¼ (4.6.28)
This is equivalent to Ptolemy’s formula for cyclic quadrilaterals (page 323).
3. In oblique coordinates with angle , a circle of center ´Ü ¼ Ý ¼ µ and radius Ö is
described by the equation
´Ü Ü ¼ µ ¾ ·´Ý Ý ¼ µ ¾ ·¾´Ü Ü ¼ µ´Ý Ý ¼ µÓ× Ö ¾ (4.6.29)
4. In polar coordinates, the equation for a circle centered at the pole and having
radius is Ö . The equation for a circle of radius passing through the
pole and with center at the point ´Öµ´ ¼ µ is Ö ¾ Ó×´ ¼ µ. The
equation for a circle of radius and with center at the point ´Öµ´Ö ¼ ¼ µ
is
Ö ¾ ¾Ö ¼ Ö Ó×´ ¼ µ·Ö ¾ ¼ ¾ ¼ (4.6.30)
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