ntre -ap +
bq + cr = 0. The circumcircle is a sqrt(p) + b sqrt(q) + c sqrt(r) =
0, the centre being p sin 2A + q sin 2B + r sin 2C = 0. The general
equation to a circle in this system of co-ordinates is deduced as
follows: If [rho] be the radius and lp + mq + nr = 0 the centre, we
have [rho] = (lp1 + mq1 + nr1)/(l + m + n), in which p1, q1, r1 is a
line distant [rho] from the point lp + mq + nr = 0. Making this
equation homogeneous by the relation [Sigma]a squared(p - q) (p - r) =
4[Delta] squared (see GEOMETRY: _Analytical_), which is generally written
{ap, bq, cr} squared = 4[Delta] squared, we obtain {ap, bq, cr} squared[rho] squared =
4[Delta] squared{(lp + mq + nr)/(l + m + n)} squared, the accents being dropped, and
p, q, r regarded as current co-ordinates. This equation, which may be
more conveniently written {ap, bq, cr} squared = ([lambda]p + [mu]q +
[nu]r) squared, obviously represents a circle, the centre being [lambda]p +
[mu]q + [nu]r = 0, and radius 2[Delta]/([lambda] + [mu] + [nu]). If we
make [lambda] = [mu] = [nu] = 0, [rho] is infinite, and we obtain {ap,
bq, cr} squared = 0 as the equation to the circular points.

_Systems of Circles._

_Centres and Circle of Similitude._--The "centres of similitude" of two
circles may be defined as the intersections of the common tangents to
the two circles, the direct common tangents giving rise to the "external
centre," the transverse tangents to the "internal centre." It may be
readily shown that the external and internal centres are the points
where the line joining the centres of the two circles is divided
externally and internally in the ratio of their radii.

The circle on the line joining the internal and external centres of
similitude as diameter is named the "circle of similitude." It may be
shown to be the locus of the vertex of the triangle which has for its
base the distance between the centres of the circles and the ratio of
the remaining sides equal to the ratio of the radii of the two circles.

With a system of three circles it is readily seen that there are six
centres of similitude, viz. two for each pair of circles, and it may be
shown that these lie three by three on four lines, named the "axes of
similitude." The collinear centres are the three sets of one external
and two internal centres, and the three external centres.

_Coaxal Circles._--A system of circles is coaxal when the locus of
points from which ta