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

156

BICIRCULAR QUARTICS.

where S' is the inverse of the circle S from which it follows that
the line ax + by = 0 has a contact of the second order with the
inverse cubic at the origin, and the latter is therefore a point of
inflexion.
233. If' three tangents be drawn to a circular cubic from the
point 0 in which the cubic cuts its asymptote, the three points of
contact will lie on a circle which passes through 0.
We have shown in ~ 92 that from a point of inflexion 0 three
tangents can be drawn to a cubic, and that the three points of
contact lie on a straight line. Hence inverting with respect to 0,
the theorem at once follows from ~ 232.
234. Every circular cubic passes through the four centres of
inversion, and also through the feet of the perpendiculars of the
triangle formed by joining any three centres of inversion.
A
F
B E 0
We have shown in ~ 213 that if A, B, C, 0 be the four centres
of inversion, any one of these points is the orthocentre of the
triangle formed by joining the other three. Also the equation of
the cubic referred to ABC is
S (la cos A + mf, cos B + ny cos C)
= I (la2 cos2 A + m/32 cos2 B + ny2 cos2 C)......(9),
where S = aa2 cos A + b32 cos B + cy2 cos C............(10),
and 1 + mn + n = 0........................(11).
Putting 3= y= 0 in (9), it follows that (9) vanishes by virtue
of (10); whence the cubic passes through A; similarly it passes
through B and C. Also since the coordinates of 0 are proportional
to sec A, sec B, sec C it follows that the cubic passes through 0.