x^2+y^2+2gx+2fy+c=0, x^2+y^2+2g'x+2f'y+c'=0.
Obviously these equations show that the curves intersect in four
points, two of which lie on the intersection of the line, 2(g - g')x +
2(f - f')y + c - c' = 0, the radical axis, with the circles, and the
other two where the lines x squared + y squared = (x + iy) (x - iy) = 0 (where i =
sqrt -1) intersect the circles. The first pair of intersections may be
either real or imaginary; we proceed to discuss the second pair.
The equation x squared + y squared = 0 denotes a pair of perpendicular imaginary
lines; it follows, therefore, that circles always intersect in two
imaginary points at infinity along these lines, and since the terms
x squared + y squared occur in the equation of every circle, it is seen that all
circles pass through two fixed points at infinity. The introduction of
these lines and points constitutes a striking achievement in geometry,
and from their association with circles they have been named the
"circular lines" and "circular points." Other names for the circular
lines are "circulars" or "isotropic lines." Since the equation to a
circle of zero radius is x squared + y squared = 0, i.e. identical with the circular
lines, it follows that this circle consists of a real point and the
two imaginary lines; conversely, the circular lines are both a pair of
lines and a circle. A further deduction from the principle of
continuity follows by considering the intersections of concentric
circles. The equations to such circles may be expressed in the form
x squared + y squared = [alpha] squared, x squared + y squared = [beta] squared. These equations show that the
circles touch where they intersect the lines x squared + y squared = 0, i.e.
concentric circles have double contact at the circular points, the
chord of contact being the line at infinity.
In various systems of triangular co-ordinates the equations to circles
specially related to the triangle of reference assume comparatively
simple forms; consequently they provide elegant algebraical
demonstrations of properties concerning a triangle and the circles
intimately associated with its geometry. In this article the equations
to the more important circles--the circumscribed, inscribed, escribed,
self-conjugate--will be given; reference should be made to the article
TRIANGLE for the consideration of other circles (nine-point, Brocard,
Lemoine, &c.); while in the article GEOMETRY: _Analytical_, the
principles of the different systems are discussed.
Trilinear co-ordinates.
The equation to the circumcircle assumes the simple form
a[beta][gamma] + b[gamma][alpha] + c[alpha][beta] = 0, the centre
being cos A, cos B, cos C. The inscribed circle is cos 1/2A sqrt([alpha])
cos 1/2B sqrt([beta]) + cos 1/2C sqrt([gamma]) = 0, with centre [alpha] =
[beta] = [gamma]; while the escribed circle opposite the angle A is
cos 1/2A sqrt(-[alpha]) + sin 1/2B sqrt([beta]) + sin 1/2C sqrt([gamma]) =
0, with centre -[alpha] = [beta] = [gamma]. The self-conjugate circle
is [alpha] squared sin 2A + [beta] squared sin 2B + [gamma] squared sin 2C = 0, or the
equivalent form a cos A [alpha] squared + b cos B [beta] squared + c cos C [gamma] squared =
0, the centre being sec A, sec B, sec C.
The general equation to the circle in trilinear co-ordinates is
readily deduced from the fact that the circle is the only curve which
intersects the line infinity in the circular points. Consider the
equation
a[beta][gamma] + b[gamma][alpha] + C[alpha][beta] + (l[alpha] +
m[beta] + n[gamma]) (a[alpha] + b[beta] + c[gamma]) = 0 (1).
This obviously represents a conic intersecting the circle
a[beta][gamma] + b[gamma][alpha] + c[alpha][beta] = 0 in points on the
common chords l[alpha] + m[beta] + n[gamma] = 0, a[alpha] + b[beta] +
c[gamma] = 0. The line l[alpha] + m[beta] + n[gamma] is the radical
axis, and since a[alpha] + b[beta] + c[gamma] = 0 is the line
infinity, it is obvious that equation (1) represents a conic passing
through the circular points, i.e. a circle. If we compare (1) with the
general equation of the second degree u[alpha] squared + v[beta] squared + w[gamma] squared
+ 2u'[beta][gamma] + 2v'[gamma][alpha] + 2w'[alpha][beta] = 0, it is
readily seen that for this equation to represent a circle we must have
-kabc = vc squared + wb squared - 2u'bc = wa squared + uc squared - 2v'ca = ub squared + va squared - 2w'ab.
Areal co-ordinates.
The corresponding equations in areal co-ordinates are readily derived
by substituting x/a, y/b, z/c for [alpha], [beta], [gamma]
respectively in the trilinear equations. The circumcircle is thus seen
to be a squaredyz + b squaredzx + c squaredxy = 0, with centre sin 2A, sin 2B, sin 2C; the
inscribed circle is sqrt(x cot 1/2A) + sqrt(y cot 1/2B) + sqrt(z cot 1/2C) =
0, with centre sin A, sin B, sin C; the escribed circle opposite the
angle A is sqrt(-x cot 1/2A) + sqrt(y tan 1/2B) + sqrt(z tan 1/2C)=0, with
centre - sin A, sin B, sin C; and the self-conjugate circle is x squared cot
A + y squared cot B + z squared cot C = 0, with centre tan A, tan B, tan C. Since in
areal co-ordinates the line infinity is represented by the equation x
+ y + z = 0 it is seen that every circle is of the form a squaredyz + b squaredzx +
c squaredxy + (lx + my + nz)(x + y + z) = 0. Comparing this equation with ux squared
+ vy squared + wz squared + 2u'yz + 2v'zx + 2w'xy = 0, we obtain as the condition
for the general equation of the second degree to represent a
circle:--
(v + w - 2u')/a squared = (w + u - 2v')/b squared = (u + v - 2w')/c squared.
Tangential co-ordinates.
In tangential (p, q, r) co-ordinates the inscribed circle has for its
equation (s - a)qr + (s - b)rp + (s - c)pq = 0, s being equal to 1/2(a +
b + c); an alternative form is qr cot 1/2A + rp cot 1/2B + pq cot 1/2C = 0;
the centre is ap + bq + cr = 0, or p sin A + q sin B + r sin C = 0.
The escribed circle opposite the angle A is -sqr + (s - c)rp + (s -
b)pq = 0 or -qr cot 1/2A + rp tan 1/2B + pq tan 1/2C = 0, with centre -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 tangents to the circles are equal is a straight line.
Consider the case of two circles, and in the first place suppose them to
intersect in two real points A and B. Then by Euclid iii. 36 it is seen
that the line joining the points A and B is the locus of the
intersection of equal tangents, for if P be any point on AB and PC and
PD the tangents to the circles, then PA.PB = PC squared = PD squared, and therefore PC
= PD. Furthermore it is seen that AB is perpendicular to the line
joining the centres, and divides it in the ratio of the squares of the
radii. The line AB is termed the "radical axis." A system coaxal with
the two given circles is readily constructed by describing circles
through the common points on the radical axis and any third point; the
minimum circle of the system is obviously that which has the common
chord of intersection for diameter, the maximum is the radical
axis--considered as a circle of infinite radius. In the case of two
non-intersecting circles it may be shown that the radical axis has the
same metrical relations to the line of centres.
[Illustration: Fig. 5]
There are several methods of constructing the radical axis in this
case. One of the simplest is: Let P and P' (fig. 5) be the points of
contact of a common tangent; drop perpendiculars PL, P'L', from P and
P' to OO', the line joining the centres, then the radical axis bisects
LL' (at X) and is perpendicular to OO'. To prove this let AB, AB1 be
the tangents from any point on the line AX. Then by Euc. i. 47, AB squared =
AO squared - OB squared = AX squared + OX squared + OP squared; and OX squared = OD squared - DX squared = OP squared + PD squared - DX squared.
Therefore AB squared = AX squared - DX squared + PD squared. Similarly AB' squared = AX squared - DX squared + DP' squared.
Since PD = PD', it follows that AB = AB'.
To construct circles coaxal with the two given circles, draw the
tangent, say XR, from X, the point where the radical axis intersects
the line of centres, to one of the given circles, and with centre X
and radius XR describe a circle. Then circles having the intersections
of tangents to this circle and the line of centres for centres, and
the lengths of the tangents as radii, are members of the coaxal
system.
In the case of non-intersecting circles, it is seen that the minimum
circles of the coaxal system are a pair of points I and I', where the
orthogonal circle to the system intersects the line of centres; these
points are named the "limiting points." In the case of a coaxal system
having real points of intersection the limiting points are imaginary.
Analytically, the Cartesian equation to a coaxal system can be written
in the form x squared + y squared + 2ax +- k squared = 0, where a varies from member to
member, while k is a constant. The radical axis is x = 0, and it may be
shown that the length of the tangent from a point (0, h) is h squared +- k squared,
i.e. it is independent of a, and therefore of any particular member of
the system. The circles intersect in real or imaginary points according
to the lower or upper sign of k squared, and the limiting points are real for
the upper sign and imaginary for the lower sign. The fundamental
properties of coaxal systems may be summarized:--