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

CIRCULAR CIUBICS.

157

To find where AC cuts the cubic, put 3 = 0, and (9) reduces to
cay (Ic cos A - na cos C) (a cos A - y cos C) = 0,
the last factor of which is the equation of BE.
235. The tangents to the cubic at the four centres of inversion
are parallel to the asymptote.
Since the four points B, E, C, F lie on the cubic and also on a
circle, it follows from ~ 225 that the line joining the third points in
which BF and EC intersect the cubic is parallel to the asymptote;
but since these lines intersect on the cubic at A, the tangent at A
is parallel to the asymptote.
Since the tangent at A is the coefficient of a2 in the equation
of the cubic, it follows from (9) and (11) that its equation is
/3 (m sin C + n cos A sin B) + y (n sin B + rn cos A sin C) = 0............ (12).
A direct proof may be given as follows. The form of (9) shows
that the line
la cos A + m/3 cos B + ny cos C = 0............(13),
or u = 0, is parallel to the asymptote. The equation of any line
parallel to (13) is u + kl = 0; and if we determine k so that this
line passes through A, we shall obtain (12), which is the tangent
at A.
236. The tangents at D, E, F intersect at a point Y on the
nine-point circle, which is common to the four triangles formed by
joining the centres of inversion.
Let the tangents at D and F intersect in Y; join EY; also
let the tangent at C cut YD, YF in M1 and N. Then since D
and F are the inverse points of C with respect to B and 0,
YDC = MCD; YFC = CF =OCM,
whence YDC + YFC = OCD = 27r-B...............(14).
Also r- DYF= YDC + YFC + CDF + CFD.........(15).
But CDF= A; CFD =CAD =C-7r............(16).
Substituting from (14) and (16) in (15) we get
DYF= 2B.