QA-Co4

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

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

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

1^st 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))

1^st 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).
These intersection points also lie on conic QA-Co4:
- QA-P1. QA-P2 ^ QA-P12. QA-P24
- QA-P1. QA-P5 ^ QA-P12. QA-P32
- QA-P1. QA-P12 ^ QA-P20. QA-P23
- QA-P1. QA-P30 ^ QA-P6 . QA-P20
- QA-P2. QA-P11 ^ QA-P4 . QA-P12
The center of QA-Co4 is QA-P29 (Complement of QA-P2 wrt the DT).
QA-Co4 is the Involutary Conjugate (see QA-Tf2) of QA-L4 (Line through QA-P1, QA-P6 and QA-P23).
The QA-DT-Conic Perspector (see QA-Co-1) of QA-Co4 is a point on the line QA-P22.QA-P29.
The infinity points of the asymptotes of QA-Co4 are the QA-Tf2-images of the intersection points of QA-L4 and QA-Co1. See [34], QFG#2196.
QA-Co4 is the QA-DT-isogonal conjugate of a line through QA-P11 parallel to QA-P4.QA-P12DT-isogonal conjugate of a line through QA-P11 parallel QA-P4.QA-P12,. See [34], QFG#2196.

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