This page is devoted to a class of cubics which solve a problem raised by Dominik Burek (ADGEOM # 579) and partially answered by Francisco Garcia Capitan
See the related paper Cubics related with concurrent tangents
Consider a fixed point P = p : q : r and a variable point Q. The circumcircle of APQ meets AB, AC again at Ab, Ac. Define Bc, Ba and Ca, Cb likewise. These six points lie on a same conic if and only if Q lies on a circular circum-conic denoted by K(P) with equation :
∑ a^4 q r y z [p (y- z) - (q -r )x] + b^2 c^2 p x [r^2 y (x+z) - q^2 z (x+y) + r (p+q) y^2 - q (p+r) z^2] = 0.
In ADGEOM #612, we find another characterization of K(P) which is the locus of Q such that the tangents at A, B, C to the circles APQ, BPQ, CPQ concur (at X). In this case, the locus of X is another circular circum-cubic X(P) passing through X(3) with equation :
∑ p^2 (r y-q z) [b^2 c^2 x^2 - 2 a^2 SA y z + c^2(a^2 - c^2) x y + b^2(a^2 - b^2) x z] = 0.
General properties of K(P)
K(P) contains :
• A, B, C, the circular points at infinity J1 and J2, P and its isogonal conjugate gP = P*,
• the infinite point of the line cP-mP where cP = complement of P and mP = midpoint(P, P*),
• the isogonal conjugate of the infinite point of the line cgP-mP, this point obviously on the circumcircle.
K(P*) is the isogonal transform of K(P) and these two cubics generate a pencil of circular circum-cubics passing through P, P* that also contains :
• the isogonal pK with pivot the infinite point of the line PP*,
• the isogonal focal nK whose root is the trilinear pole of the last common tangent of the Steiner inellipse and the inscribed conic with foci P and P*, with center mP.
All these cubics must meet at two other (real or not) isogonal conjugate points P1, P2 lying on a parallel to PP*.
Special cubics K(P)
• K(O) splits into the line at infinity and the Jerabek hyperbola.
• K(H) = K(O)* splits into the circumcircle and the Euler line.
• K(P) is a focal cubic if and only if P lies on the Napoleon cubic K005.
• K(P) is a nK if and only if P lies on a complicated tricircular isogonal sextic passing through the in/excenters, O, H.
• K(P) is a psK if and only if P lies on the line at infinity (1) or on the circumcircle (2). More precisely,
- (1) K(P) = psK(P x X6, X2, X3) contains O, the midpoints of ABC. The singular focus lies on C(O, R/2) and the pseudo-pole lies on the Lemoine axis.
- (2) K(P) = psK(P, P x X76, X4) contains H and is tangent at A, B, C to the symmedians. The singular focus lies on C(O, 2R) and the pseudo-pivot lies on the Steiner ellipse. In this case, K(P) is the locus of the pivots of circular pKs passing through P.
General properties of X(P)
X(P) contains :
• A, B, C, the circular points at infinity J1 and J2, O, P, igP = inverse in the circumcircle of gP,
• the infinite point of the line P-H/P where H/P is the H-Ceva conjugate of P,
• the isogonal conjugate of the infinite point of the line P-gP, this point obviously on the circumcircle.
Special cubics X(P)
• X(H) splits into the line at infinity and the Jerabek hyperbola.
• X(I) splits into the circumcircle and the line OI.
• X(P) is a focal cubic if and only if P lies on the Napoleon cubic K005.
• X(P) is a pK if and only if P lies on the second Brocard cubic K018.
• the polar conic of O is a rectangular hyperbola if and only if P lies on the Neuberg cubic K001.
• when P lies on the line at infinity, the cubics X(P) and K(P) coincide.
The following table gives a selection of such cubics K(P)