QA-L8: QA-DT-Isogonal Conjugate of QA-Co1

 
QA-L8 is the line which is the Isogonal Conjugate of the Nine-point Conic QA-Co1 wrt the QA-Diagonal Triangle. This line is mentioned with several properties by Benedetto Scimemi in Ref-36, page 347. It is also used for the construction of the axes / asymptotes of QA-Co1. See also QA-Co1.
 
 
QA-L8-IsogonalConugate-of-QA-Co1-01
 
 
Coefficients:                     
1st CT-coefficient QA-L8:
                c4 p q2 (p + r) (q + r) (-2 q r2 + p2 (-q + r) + p r (q + r)) + (p + q) r2 (a4 q2 (p + r) (p2 - 3 q r + p (q + r))
+ b4 p (q + r) (p2 (q - r) - 2 q2 r + p q (q + r)) - 2 a2 b2 p q (p2 q + q r (-2 q + r) + p (q + r)2))
+ 2 c2 p q r (-a2 q (p + r) (p2 r + q (q - 2 r) r + p (q + r)2) + b2 (q + r) (2 q2 r2 + p2 (q2 + r2)))
 
1st CT-coordinate Infinity point QA-L8:
                -p (q + r) (c4 p2 q (p + r) (q2 - r2) + c2 q r (2 b2 p2 (-q2 + r2) + a2 (p + r) (-p2 (q - 2 r) + q2 r + p q (q + r)))
+ (p + q) r (a4 p q (q - r) (p + r) + b4 p2 (q2 - r2) - a2 b2 q (p2 (2 q - r) + q r2 + p r (q + r))))
1st DT-coefficient QA-L8:
p2/a2
 
1st DT-coordinate Infinity point QA-L8:
            a2 (b r + c q) (b r - c q)
 
Properties:
  • The QA-DT-Isogonal Conjugate of QA-P3 is the projection point of QA-P11 on QA-L8. See Ref-36, 3.4.3 and 3.4.4 at page 347. It is also the Complement of QA-P4 wrt QA-DT.
  • The QA-DT-Isogonal Conjugate of QA-P2 is the infinity point of QA-L8. See Ref-36, 3.4.3 and 3.4.4 at page 347.
  • The reflections of QA-P2 in the sides of QA-DT are collinear on the line QA-P4.QA-P12, which is a line perpendicular to QA-L8. See Ref-36, 3.4.3 and 3.4.4 at page 347.
  • The line QA-P13.QA-P28 is also perpendicular to QA-L8.
  • The QA-DT Simson Line of QA-P2 being QA-P6.QA-P36 is also perpendicular to QA-L8. (Eckart Schmidt in Ref-34, QFG # 424).
  • QL-L8 is the Trilinear Polar of QA-P16*, the QA-DT-Isogonal Conjugate of QA-P16. (Eckart Schmidt in Ref-34, QFG # 424).


 

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