QA-Co4: QA-DT-P3-P12 Orthogonal Hyperbola

QA-Co4 is the circumscribed orthogonal hyperbola of the Diagonal Triangle of the Reference Quadrangle passing through the Gergonne-Steiner Point (QA-P3).

Equations:

Conic-equation in CT-notation:
(a2 (q - r)     /(q + r) - b2 (p + 3 r)/(p + r) + c2 (p + 3 q)/(p + q)) q r x2
+ (a2 (q + 3 r) /(q + r) - b2 (p - r)     /(p + r) - c2 (3 p + q) /(p + q)) p r y2
+ (-a2 (3 q + r)/(q + r) + b2 (3 p + r)/(p + r) + c2 (p - q)     /(p + q)) p q z2
- (a2 (p q + q2 - p r + 3 q r)/(q + r) - b2 (p2 + p q + 3 p r - q r)/(p + r) + c2 (p - q)) r x y
- (a2 (p q - p r - 3 q r - r2)   /(q + r) - b2 (p - r) + c2 (p2 + 3 p q + p r - q r)/(p + q)) q x z
- (a2 (q - r) - b2 (p q - 3 p r - q r - r2)/(p + r) -   c2 (3 p q + q2 - p r + q r)/(p + q)) p y z = 0
Conic-equation in DT-notation:
((a2-b2) r2 - c2 (p2-q2)) x y
+ ((c2-a2) q2 - b2 (r2-p2)) x z
+ ((b2-c2) p2 - a2 (q2-r2)) y z = 0

1st CT-coordinate QA-DT-Conic Perspector (see QA-Co-1):
(a2 (p + q) (q - r) (p + r) (2 p + q + r) - b2 (p + q) (q + r) (2 p2 + p q + 3 p r - q r - r2) + c2 (p + r) (q + r) (2 p2 + 3 p q - q2 + p r - q r)) /
(a2 (p + q) (q - r) (p + r) - b2 (p + q) (q + r) (p + 3 r) + c2 (p + 3 q) (p + r) (q + r))
1st DT-Coordinate QA-DT-Conic Perspector (see QA-Co-1):
(b2 p2 - c2 p2 - a2 q2 + a2 r2)
(c4 p2 (p - q) q2 (p + q) (p2 - q2 - r2) + b4 p2 (p - r) r2 (p + r) (p2 - q2 - r2) + 2 b2 c2 p2 q2 r2 (3 p2 - q2 - r2) + a4 q2 r2 (p2 q2 - q4 + p2 r2 + 2 q2 r2 - r4) + a2 (-2 c2 p2 q2 r2 (p2 + q2 - r2) - 2 b2 p2 q2 r2 (p2 - q2 + r2)))

Properties:
• QA-Co4 passes apart from the vertices of the Diagonal Triangle also through:
QA-P3 (Gergonne-Steiner Point),
QA-P12 (Orthocenter QA-Diagonal Triangle),
QA-P20 (Reflection of QA-P5 in QA-P1)
QA-P30 (Reflection of QA-P2 in QA-P11).