The equation to the circumcircle assumes the simple form a fry +bra+ca(3= o, thecentre being cos A, cos B, cos C. The inscribed circle is cos zA -V a +cos 1B -J (3 +cos 2C ¦ y = o, with centre Trill ea a= (3 = y; while the escribed circle opposite the angle A is cos 2A' - a+sin 2B A / 0+sin IC y=o, with centre Hates.
The circumcircle is thus seen Areal to be a 2 yz+b 2 zx+c 2 xy=o, with centre sin 2A, sin 2B, co sin 2C; the inscribed circle is A t (x cot ZA)+ (y cot 2B) nates.
The circumcircle is all q+c -s1 r = o, the centre being sin 2A+q sin 2B+r sin 2C =o.
Duced by euclidian methods from the definition include the following: the tangent at any point bisects the angle between the focal distance and the perpendicular on the directrix and is equally inclined to the focal distance and the axis; tangents at the extremities of a focal chord intersect at right angles on the directrix, and as a corollary we have that the locus of the intersection of tangents at right angles is the directrix; the circumcircle of a triangle circumscribing a parabola passes through the focus; the subtangent is equal to twice the abscissa of the point of contact; the subnormal is constant and equals the semilatus rectum; and the radius of curvature at a point P is 2 (FP) 4 /a 2 where a is the semilatus rectum and FP the focal distance of P.
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").
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