QA-L4: QA-P1-P6 Line

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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).

                      

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))

1^st 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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