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

THE FOCAL PARABOLA.

159

Since by ~ 216 the triangle is self-conjugate to the parabola,
the equation of the latter may be written
lac + m 2 + ny2 = 0..................... (17),
where a2/1 + b2/m + c2/n = 0.................... (18).
Let D1, El and F1 be the middle points of BC, CA, AB; then
the equation of DE, is
a2 + b - cy = 0....................... (19),
and (18) is the condition that (19) should touch (17). Hence the
parabola touches the sides of the triangle DiE1F, and therefore its
focus lies on the circle circumscribing D1,EYF, that is upon the
nine-point circle of ABC.
240. The directrices of the four focal parabolas pass respectively through the centres of the four circles circumscribing the four
triangles formed by joining the centres of inversion.
The equation of the directrix of (17) is*
I (m + n) a/a + m (n + I) s/b + nit (1 m) y/c = 0.
The condition that this line should pass through the centre of the
circle circumscribing ABC, whose coordinates are proportional to
cos A, cos B and cos C, is
Im (cot A + cot B) + mn (cot B + cot C) + nl (cot C+ cot A) = 0,
which is the same thing as (18), which is the condition that (17)
should be a parabola.
On the Points of Infiexion.
241. The general equation of a circular cubic may be written
in the form
Su = Iu2.......................(20),
where S is the circumscribing circle, and
u. = Xa + / 93 + v +....................................... (21),
u = la2 + m/32 + ny2 + 21'/3y + 2m'ya + 2n'a/...... (22).

* Ferrers' Trilinear Coordinates, p. 93.