QA-P29: Complement of QA-P2 wrt the Diagonal Triangle


QA-P29 is the complement of QA-P2 (Euler-Poncelet Point) wrt the Diagonal Triangle.
                       
QA-P29-Complement of QA-P2 wrt DT-00 
 
Coordinates: 
1st CT-Coordinate:
-2 a4 p q r (p + q) (p + r) (q + r) (2 p + q + r)
- b4 p r (p + q) (q + r)2 (3 p2 + p q + p r - q r)
- c4 p q (p + r) (q + r)2 (3 p2 + p q + p r - q r)
- b2 c2 p   (q + r)2 (p3 q + p2 q2 + p3 r - 4 p2 q r - p q2 r + p2 r2 - p q r2 + 2 q2 r2)
+ c2 a2 p q (p + r) (q + r) (p2 q + p q2 + 7 p2 r + 4 p q r + q2 r + 3 p r2 - q r2)
+ a2 b2 p r (p + q) (q + r) (7 p2 q + 3 p q2 + p2 r + 4 p q r - q2 r + p r2 + q r2)
 
1st DT-Coordinate:
(c2 q2 - b2 r2) ((c2 - b2) p2 + a2 (q2 - r2))
 
Properties:
  • QA-P29 lies on these QA-lines:
        QA-P2.QA-P10          (3 : -1 => QA-P29 = Complement QA-P2 wrt QA-DT)
        QA-P3.QA-P20          (1 : 1 => QA-P29 = Midpoint QA-P3.QA-P20)
        QA-P5.QA-P34          (3 : -1)
        QA-P6.QA-P28          (2 : -1 => QA-P29 = Reflection of QA-P6 in QA-P28)
        QA-P12.QA-P30        ( 1 : 1 => QA-P29 = Midpoint)
        QA-P13.QA-P36        (-1 : 2 => QA-P29 = Reflection of QA-P36 in QA-P13)
  • QA-P29 is the center of the QA-DT-circumscribed orthogonal hyperbola (QA-Co4) additional passing through QA-P3, QA-P12, QA-P20, QA-P30.
  • QA-P29 lies on the Medial Circle of the QA-Diagonal Triangle (QA-Ci2).
  • QA-P29 lies on the Simson Line of QA-P30 occurring on the circumcircle of the QA-Diagonal Triangle (QA-Ci1).
  • QA-P29 = QA-P1 (QA-Centroid) of the Quadrangle QA-P3.QA-P12.QA-P20.QA-P30.
  • QA-P29 = QA-P2 (Euler-Poncelet Point) of the quadrangle formed by the vertices of the QA-Diagonal Triangle and QA-P3 of the Reference Quadrangle.
  • QA-P29 = QA-P1 (QA-Centroid) of the Quadrangle formed by the DT-vertices + Reflection of QA-P2 in QA-P29. 
  • If we consider for the QL-inscribed conics the contact QA, their QA-P29 will lie on QL-Ci2. See Ref-66, Eckart Schmidt, QPG-message #1155.