An elementary treatise on cubic and quartic curves, by A. B. Basset.

142

BICIRCULAR QUARTICS.

It is known from the geometry of the circle, that the radical
axes of any three circles intersect in a point which is called the
radical centre of the three circles; and that the tangents drawn
from the radical centre to each of the three circles are equal.
Hence the circle whose centre is the radical centre and whose
radius is equal to any one of the tangents to the three circles cuts
each of them orthogonally; also if any number of circles have a
common radical centre, a circle can be described cutting each of
them orthogonally.
Let S be the circle circumscribing the triangle of reference;
then we may write
U = S - (11a + m,3 + ny) I,
with similar expressions for V and W. Whence the radical axes
of U and V, V and W, W and U are
(z, - 1) a + (i, - min) 3 + (n, - 2) 7 = 0, &c., &c.
The radical axis of U and (19) is
{a ( - 12) - (13 - 1)} a + I{ (Mi - n2) - v (m3 - m1)} l3
+ {/, (n - 'n2) - v (n3 - n1)} 7 = 0,
which obviously passes through the radical centre of U, V and TV.
Hence the circle which cuts U, V, W orthogonally cuts (19)
orthogonally. This circle is therefore the circle of inversion, the
circle (19) is the generating circle, whilst the conic (22) is the
focal conic.
211. It is shown in treatises on Conics, that if a circle and a
conic intersect in four points P, Q, R, S; and if SP, RQ intersect
in A; PR, QS in B; and PQ, SR in C; the triangle ABC is
self-conjugate to the conic and the circle, and the orthocentre of
ABC is the centre of the circle. If therefore the radii of the
circles U, V, W be chosen so that the orthocentre is their radical
centre, the circle through P, Q, R, S will cut (19) orthogonally.
Accordingly the former circle is the fixed circle or circle of inversion, whilst (19) is the generating circle; hence the quartic may
be generated in a third manner:Let the focal conic cut the circle of inversion in P, Q, R, S;
let SP, QR intersect in A; PR, SQ in B; PQ, SR in C. With
A, B, C as centres describe three circles U, V, W, whose radii are
such that the orthocentre of ABC is their radical centre; then the