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

154

BICIRCULAR QUARTICS.

If S be taken as the circle circumscribing ABC, the conic u2
must also circumscribe this triangle, whence
Ioo = l3,y + myca + na3,
l, = Xa + -L3- + v-y.
Let the lines AB, CD meet the cubic in E and F; then since
D is the fourth point in which S and u, intersect, the equation
of CD is
c (3 + ma)= n (a3 + ba)..................(7).
To find the third point F where CD intersects the cubic,
substitute the left-hand side of (7) in the term u, in the cubic and
it reduces to
(cul - nI) S = 0,
which shows that the line
c l - nI = O...........................(8)
passes through F. Putting 7 = 0 in (8) and also in (6) it follows
that (8) passes through E; whence (8) is the equation of EF.
The form of (8) shows that EF is parallel to the asymptote.
226. If A and B coincide, AE is the tangent at A, whence:If a circle touch the cubic at A and intersect it at C and D, the
tangent at A and the line CD intersect the cubic at two points E
and F, such that EF is parallel to the asymptote.
227. Let C and D as well as A and B coincide, then:If a circle touch the cubic at two points, the line joining the two
points, where the tangents at the points of contact cut the cubic, is
parallel to the asymptote.
228. Let A, B and C coincide, then:If the chord of curvature intersect the cubic in F, and the
tangent to the cubic and its circle of curvature meet the curve in E,
the line EF is parallel to the asymptote.
229. If a straight line parallel to the asymptote of a circular
cubic cut the curve in a and c, and if the tangents at a and c cut
the curve in A and C, then the four points AacC lie on a circle;
also AC intersects the cubic at the point where it is cut by its
asymptote.