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

THE CARDIOID.

199

cusps at the circular points at infinity. Also the curve is of the
third class. The curve accordingly belongs to the tenth division
of quartics.
301. Cuspidal chords are of constant length; and tangents at
their extremities intersect at right angles on a fixed circle, whose
centre is the triple focus.
TK Aq bs e owA
Let F be the cusp, C the centre of the circle of which the
cardioid is the pedal; Qq any diameter of the circle. Then if Pp
be drawn through F parallel to Qq cutting the tangents at Q and
q to the circle in P and p; Pp is a cuspidal chord of the cardioid,
and is equal to Qq.
Let the tangents at P, p intersect in T; then
TPF= FQP,
TpP = Fqp = FQq = 7 - FQP,
whence TPp + TpP = ~ r,
and therefore PTp is a right angle. Also the triangles QFq and
pTP are equal in every respect.
Join At, Tt, where A is the other extremity of the diameter
FC; then tp = FP, Tp = FQ, also TpP = PFQ, whence the triangles
Tpt and QFP are equal in every respect, and therefore Ttp = r,
and TtA is a straight line.
Draw TK perpendicular to AT; then since Tt = FR = At;
AK =-AF, and therefore K is a fixed point. The locus is therefore a circle whose centre E is such that FE = IFA.