QA-L4: QA-P1-P6 Line


The QA-L4-Line is the line through the Midpoint of the segment (QA-P2,QA-P3) on QA-L1 and the Midpoint of the segment (QA-P2,QA-P4) on QA-L2, being QA-P1 (Centroid) and QA-P6 (Parabola Axes Crosspoint).
QA-L4--P1-P6-Line                      
Coefficients: 
1st CT-coefficient:
a2 (q - r)/(q + r) - b2 (p + 3 r)/(p + r) + c2 (p + 3 q)/(p + q)
1st DT-coefficient:
            q2 r2 ((-b2+c2) p2+a2 (q2-r2))
1st CT-Coordinate Infinity Point:
a2 - b2 p/(p + r) - c2 p/(p + q)
1st DT- Coordinate Infinity Point:
            p2 (c2 q2 (p2 - q2 + r2) + r2 (-2 a2 q2 + b2 (p2 + q2 - r2)))
Properties:
  • QA-L4 // QA-P3.QA-P4 // QG-P1.QG-P16.
  • QA-L4 // QA-Cu1-asymptote.
  • Also QA-P23 (Inscribed Square Axes Crosspoint) lies on QA-L4.
  • The two asymptotes of the Orthogonal Quadrangle Hyperbola (QA-Co2) and the two axes of The Gergonne-Steiner Conic (QA-Co3) form a rectangle. QA-P2 and QA-P3 are 2 opposite vertices of this rectangle. The other 2 vertices lie on QA-L4.
  • QA-Co4 is the Involutary Conjugate (see QA-Tf2) of QA-L4.
  • The Infinity Point of QA-L4 = the Involutary Conjugate (see QA-Tf2) of QA-P3.
  • The 3 QA-versions of QL-P5 are collinear on a line parallel to QA-L4 (note Eckart Schmidt).


 

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