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

THE CARDIOID.

201

Eliminating cos z between (4) and (5) we obtain
8 (X24 y2 - b2)2 + 6b2 (x + y2 - b) + 3b3 (2 - b) = 0......(6).
Transfer the origin to the point x=- b, and the equation
reduces to
4 (x2 + y2)2 + b (2 + y2) (8-:3b) + 4 b2X2 = 0,
or r = b (/3 - 2 cos 0).
303. The angular points of a given triangle move round the
circumference of a fixed circle; prove that the directrices of the
system of parabolas which have a given focus and touch the three
sides of the triangle envelope a cardioid.
K
S A
B M /
Let ABC be the triangle, 0 its orthocentre, I the centre of
the circle, S the focus of the system of parabolas. Then it is
known from the geometry of the parabola (i) that S lies on the
circle, (ii) that the directrix of every parabola which touches the
sides of the triangle passes through the orthocentre, (iii) the pedal
line of S is the tangent at the vertex of the parabola; hence the
directrix is parallel to the pedal line KLM.
Draw IZ perpendicular to the directrix, and let SIZ=,
MKC=. Then
p = IZ= 2SY+ Rcos#,
and SY= R (cos B + cos ISK) cos;
also ISK = IAS + ASK
= IAL + SK K + KLA
=7r-2(b- C+ A,
and ISK = t - 4 + b,

whence

*r = 3~ + C-A.