QG-L2: The Harmonic Line


The Harmonic Line is the line through QA-P16 (the harmonic point of a Quadrangle) and QL-P13 (the harmonic point of a Quadrilateral) both meeting in their overlap of a Quadrigon.
 QG-L2-Harmonic Line-00
 

Coefficients: 
CT-Coefficients QG-L2 in 3 QA-Quadrigons:
  • (q r (-q + r) : p r (2 p + q + r) :   -p q (2 p + q + r))
  • (q r (p + 2 q + r) :   p r (r - p)  :   -p q (p + 2 q + r))
  • (q r (p + q + 2 r) :  -p r (p + q + 2 r) :  p q (-p + q))
CT-Coefficients QG-L2 in 3 QL-Quadrigons:
  • ( 2 l2 (m - n) : m (l m - l n + m n) : -n (-l m + l n + m n) )
  • ( l (l m + l n - m n) : 2 m2 (l - n)   : -n (-l m + l n + m n) )
  • ( l (l m + l n - m n) : -m (l m - l n + m n) :   2 (l - m) n2 )
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DT-Coefficients QG-L2 in 3 QA-Quadrigons:
  • ( r2 : 0 : -p2)
  • ( 0 : -r2 : q2)
  • (-q2 : p2 : 0 )
DT-Coefficients QG-L2 in 3 QL-Quadrigons:
  • ( l2 :  0  : -n2)
  • ( 0 : -m2 : n2)
  • (-l2 : m2 :  0 )

Properties:
  • QA-P16, QL-P13 are collinear with QG-P1, QG-P12 and QG-P13.
  • QG-L2 is the radical axis of the circumcircle of the QA-Diagonal Triangle and the circumcircle of the QL-Diagonal Triangle.
  • Let T be the intersection point QG-L1^QG-L2.
           Now (QG-P1,T) and (QG-P12,QA-P16) are harmonic conjugated pairs on QG-L2.
           Also (QG-P1,T) and (QG-P13,QL-P13) are harmonic conjugated pairs on QG-L2.
           T is also the Involution Center of (QL-DT1,QL-DT2) and (QA-DT1,QA-DT2).
           (notes Eckart Schmidt)