QA-Co3: Gergonne-Steiner Conic

In a random quadrangle it is generally impossible to circumscribe a circle through all 4 points. But it is always possible to circumscribe a conic (ellipse, parabola or hyperbola) through these 4 points. Technically, the conic with least eccentricity is the conic that deviates least from a circle.
The Gergonne-Steiner Conic is the conic through the vertices of the Reference Quadrangle with least eccentricity. It is the conic with center QA-P3 (Gergonne-Steiner Point). The problem of the Quadrangle conic with least eccentricity was initially posed in the “Annales de Gergonne” and solved by J. Steiner. See Ref-7 page 231. Equations:
Equation CT-notation:
(a2 p q r (2p+q+r) - (b2 r +c2 q) p2 (q+r)) y z
+ (b2 p q r (p+2q+r) - (c2 p +a2 r) q2 (p+r)) z x
+ (c2 p q r (p+q+2r) - (a2 q+b2 p) r2 (p+q)) x y = 0
Equation DT-notation:
(2 a2 q2 r2 - b2 r2 (p2+q2-r2) - c2 q2 ( p2 - q2+r2)) x2
+(2 b2 p2 r2 - a2 r2 (p2+q2-r2) - c2 p2 (-p2+q2+r2)) y2
+(2 c2 p2 q2 - a2 q2 (p2-q2+r2) - b2 p2 (-p2+q2+r2)) z2 = 0

Properties:
• Jean-Pierre Ehrmann commented at Hyacinthos wrt this conic that when e = eccentricity,
then for this conic e2 = [2/(k+1)],
where k = O1P1*/R1 = O2P2*/R2 = O3P3*/R3 = O4P4*/R4,
Oi = circumcenter PjPkPl,
Pi*= isogonal conjugate of Pi wrt PjPkPl,