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

144

BICIRCULAR QUARTICS.

the two foci which coincide at R must be real foci. We thus
obtain the following theorem:When a bicircular quartic has a real node, the latter arises from
the union of two real single foci; and when it has a real cusp, the
latter arises from the union of three real single foci. We shall
have examples of this in the case of the lima9on and the cardioid.
When the circle of inversion has a double contact with the
focal conic, each point of contact will be a double point on the
quartic, which together with the circular points at infinity make
four double points. Since this is greater than the maximum
number, the quartic must break up into two conies each of which
passes through the circular points at infinity, and must therefore
be circles.
213. Before proceeding further with the theory of bicircular
quartics, it will be desirable to consider certain geometrical propositions connected with the circle.
Let ABC be any triangle, 0 its orthocentre; then
(i) The triangle formed by joining any three of the four points
A, B, C and 0 has the fourth point for its orthocentre.
(ii) The four triangles thus formed have a common nine-point
circle.
For the points D, E, F are the feet of the perpendiculars
drawn from the angles of each of the four
A triangles on to the opposite sides; and the
nine-point circle is the circle circumscribing
the triangle DEF.
pa^/ f (iii) Each point is the centre of the circle
to which the triangle formed by joining the
Dc remaining three is self-conjugate.
B D
(iv) The four circles, to which the four
E s triangles are self-conjugate, cut one another
orthogonally.
Let 8, r1, r,, r3 be the radii of the four circles to which the
triangles ABC, OBC, OCA and OAB are respectively self-conjugate. Then since A is the pole of BC with respect to the circle
to which the triangle ABC is self-conjugate,

= OD. OA =- 4R2 cos A cos B cos C,