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.
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:
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QG-P13 lies on these lines:
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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 Involutary Conjugate (QA-Tf2) of QG-P13 is the infinity Point of the line QG-P1.QG-P2.