QA-Co2: QA-Orthogonal Hyperbola


Since an Orthogonal Hyperbola (OH) is determined by 4 points there is only one Orthogonal Hyperbola defined by the vertices of a quadrangle.
QA-Co2 is the unique Orthogonal Hyperbola defined by the vertices of the reference quadrangle.
 QA-Co2-OrthogonalHyperbola-00

Equations:
Equation CT-notation:
            (SA x SB y) p q z + (SB y SC z) q r x + (SC z - SA x) p r y = 0
Equation DT-notation:
            (b2 r2 - c2 q2) x2 + (c2 p2 - a2 r2) y2   (b2 p2 - a2 q2) z2 = 0

Properties:
  • The orthocenters of all QA-Component Triangles lie on this hyperbola.
  • QA-P2 (Euler-Poncelet Point) is the center of the QA-Orthogonal Hyperbola.
  • The asymptotes of the QA-Orthogonal Hyperbola are parallel to:
        the axes of the Nine-point Conic (QA-Co1)
        the axes of the Gergonne-Steiner Conic (QA-Co3).
        the reflection axes of the QA-Orthopole Transformation (QA-Tf3).
  • The incenter and the excenters of the QA-Diagonal Triangle (QA-Tr1) lie on QA-Co2. This property can be found in Ref-28, page 100, theorem 4.2. This property can be found at Ref-36 page 347, as well as at Ref-28, page 100, theorem 4.2.  See also Ref-11, Hyacinthos message # 21130 of Floor van Lamoen.
  • The QA-Diagonal Triangle (QA-Tr1) is self-polar wrt QA-Co2.
  • QA-P11.QA-P41 intersects QA-Co2 on QA-Cu1. See Ref-34, Eckart Schmidt, QFG-message #1666.
  • The vertices of the anticevian triangle (wrt QA-Tr1) of any point of QA-Co2 also will be resident on QA-Co1. See Ref-34, Eckart Schmidt, QFG-message #1666.
  • Let P be on QA-Co2 and X, Y its reflections in PiPj and PkPl, then the circumcircles of XPiPj and YPkPl intersect in the reflection of P in QA-P2. See Ref-34, Eckart Schmidt, QFG-message #1666.