The self-conjugate circle is a 2 sin 2A +0 2 sin 2 B +y 2 sin 2C = o, or the equivalent form a cosAa 2 +bcosB(2 +ccosCy 2 = o, the centre being sec A, sec B, sec C.
It also follows that a line half-way between a point and its polar and parallel to the latter touches the parabola, and therefore the lines joining the middle points of the sides of a self-conjugate triangle form a circumscribing triangle, and also that the ninepoint circle of a self-conjugate triangle passes through the focus.
Try = o to be a parabola is lbc+mca+nab = o, and the conic for which the triangle of reference is self-conjugate la 2 +143 2 +n7 2 =o is a 2 inn--+b 2 nl+c 2 lm=o.
In this article the equations to the more important circles - the circumscribed, inscribed, escribed, self-conjugate--will be given; reference should be made to the article Triangle for the consideration of other circles (nine-point, Brocard, Lemoine, &c.); while in the article Geometry: Analytical, the principles of the different systems are discussed.
The orthocentre of a triangle circumscribing a parabola is on the directrix; a deduction from this theorem is that the centre of the circumcircle of a self-conjugate triangle is on the directrix ("Steiner's Theorem").