QG-P13: Circumscribed Harmonic Conic Center

 
The Circumscribed Harmonic Conic Center is the Center of the circumscribed Harmonic Conic QG-Co2. This conic touches the sidelines of the projective circumscribed Quadrigon in the vertices of the Reference Quadrigon.
See picture below.
 QG-P13-Circumscr-HarmonicCenter-02
Coordinates:
 
CT-Coordinates QG-P13 in 3 QA-Quadrigons:
  • (2p (2p+q+r) :     q (2p+q-r)     :   r (2p - q+r))
  • ( p (p+2q -r)   : 2q (p+2q+r)   :   r (-p+2q+r))
  • ( p (p -q+2r)   :     q (-p+q+2r) : 2r ( p+q+2r))
 
CT-Coordinates QG-P13 in 3 QL-Quadrigons:
  • (m n     (l m + l n + m n) : l n (-3 l m + l n + m n) : l m (l m - 3 l n + m n))
  • (m n (-3 l m + l n + m n) : l n     (l m + l n + m n) : l m (l m + l n - 3 m n))
  • (m n   (l m - 3 l n + m n) : l n   (l m + l n - 3 m n) : l m   (l m + l n + m n))
 
CT-Area of QG-P13–Triangle in the QA-environment:
  • 54 p q r (p + q) (p + r) (q + r) Δ     /    (((q - r)2 + 4 p (p + q + r)) ((p - r)2 + 4 q (p + q + r)) ((p - q)2 + 4 r (p + q + r)))
 
CT-Area of QG-P13–Triangle in the QL-environment:
  • 16 l2 m2 n2 (lm-ln-mn) (lm+ln-mn) (lm-ln+mn) Δ
/ ((l2m2+l2n2+m2n2+2lmn(l+m-3n)) (l2m2+l2n2+m2n2+2lmn(l-3m+n))  (l2m2+l2n2+m2n22lmn(-3l+m+n)))
 

 
DT-Coordinates QG-P13 in 3 QA-Quadrigons:
  • ( - p2 : 2q2 : 2 r2)
  • (2 p2 :  -q2 : 2 r2)
  • (2 p2 : 2q2 : - r2)
 
DT-Coordinates QG-P13 in 3 QL-Quadrigons:
  • (-m2 n2 :  l2 n2 : l2 m2)
  • ( m2 n2 : -l2 n2 :  l2 m2)
  • ( m2 n2 : l2 n2 : -l2 m2)
 
DT-Area of QG-P13–Triangle in the QA-environment:
  • -27 S p2 q2 r2 / (2 (-p2+2 q2+2 r2) (2 p2+2 q2-r2) (2 p2-q2+2 r2))
 
DT-Area of QG-P13–Triangle in the QL-environment:
  • -2 S l4 m4 n4 / ((-l2 m2+l2 n2+m2 n2) (l2 m2+l2 n2-m2 n2) (l2 m2-l2 n2+m2 n2))
 

 
Properties:
        QG-P1.QG-P12 (=QG-L2)
        QG-P2.QG-P14
        QA-P3.QL-P17
  • QG-P13 is the intersection of lines from an intersection of a diagonal with the 3rd diagonal and midpoint of the other diagonal (note Eckart Schmidt).
  • The triangle formed by the 3 QA-versions of QG-P13 is perspective with the triangle formed by the 3 QA-versions of QG-P1 (the QA-Diagonal Triangle). Their Perspector is QA-P16 (QA-Harmonic Center).
  • The triangle formed by the 3 QL-versions of QG-P13 is perspective with the QL-Diagonal Triangle. Their Perspector is QL-P13 (QL-Harmonic Center).
  • The circumcenter of the triangle formed by the 3 QL-versions of QG-P13 is a point on QL-L6 (Quasi Ortholine).
  • QG-P13 lies on the Nine-point Conic QA-Co1. So the vertices of the triangle formed by the 3 QA-versions of QG-P13 lie on the Nine-point Conic QA-Co1.
  • The Involutary Conjugate (QA-Tf2) of QG-P13 is the infinity Point of the line QG-P1.QG-P2.
  • QG-P13= the QL-DT-Trilinear Pole of the QA-DT-Trilinear Polar of QG-P14 (Eckart Schmidt, October 13, 2012). For definition Trilinear Pole and Polar see Ref-13.

 
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