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.

Coefficients:
• (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))
• ( 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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• ( r2 : 0 : -p2)
• ( 0 : -r2 : q2)
• (-q2 : p2 : 0 )
• ( 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)

Vernieuwen