QA-Co1: Nine-point Conic


The Nine-point Conic is the conic through the midpoints of all possible line segments connecting the vertices of a Quadrangle.
Apart from these 6 midpoints it also passes through the 3 intersection points of all possible pairs of lines connecting the vertices of the Quadrangle. This gives a total of 9 points.
Moreover, there is a analogy with the Nine-point Circle of a Triangle that also passes through the midpoints of all possible line segments connecting its vertices.
 
QA-Co1-Ninepoint-Conic-00
 
Construction of Axes/Asymptotes of QA-Co1:

QA-Co1-Asymptotes-Axes-construction-01
 
 
Equations/infinity points:
Equation CT-notation:
            q r x2 + p r y2 + p q z2 - r (p + q) x y - q (p + r) x z - p (q + r) y z = 0
Equation DT-notation:
            r2 x y + q2 x z + p2 y z = 0
Infinity points CT-notation:
            ( p (q + r) : p q √(p q r (p + q + r)) : p r + √(p q r (p + q + r)) )
            ( p (q + r) : p q + √(p q r (p + q + r)) : p r √(p q r (p + q + r)) )
Infinity points DT-notation:
            (2 p2 : -p2-q2+r2-√(-4 p2 q2+(-p2-q2+r2)2) : -p2+q2-r2+√(-4 p2 q2+(-p2-q2+r2)2) )
            (2 p2 : -p2-q2+r2+√(-4 p2 q2+(-p2-q2+r2)2) : -p2+q2-r2-√ (-4 p2 q2+(-p2-q2+r2)2) )
These infinity points are equal to the infinity points of the 2 Quadrangle Parabolas.

Properties:
  • The center of each circumscribed conic through the vertices of the Reference Quadrangle lies on the Nine-point Conic.
  • QA-Co1 is the Involutary Conjugate of the Line at Infinity.
  • The asymptotes of the Nine-point Conic are parallel to
        the axes of the QA-parabolas (provided they are constructible).
  • The axes of the Nine-point Conic are parallel to
        the asymptotes of the QA-Orthogonal Hyperbola (QA-Co2) as well as
        the axes of the Gergonne-Steiner Conic (QA-Co3).
        the reflection axes of the QA-Orthopole Transformation (QA-Tf3).
  • Let α be the angle formed by the asymptotes of QA-Co1. The homothety coefficient of QA-Tf3 equals 2 cos α (this holds for convex quadrangles; a similar formula holds for the non-convex case). See Ref-36, pages 348, 349.
  • The line QG-P1.QG-P3 is tangent at QG-P1 to QA-Co1.
  • QA-P1 (Quadri Centroid) is the center of QA-Co1.
  • The QA-DT-Conic-Perspector (see QA-Co-1) of QA-Co1 is QA-P1.
  • QA-P2 (Euler-Poncelet Point), QA-P3 (Gergonne-Steiner Point) lie on QA-Co1.
  • The Quadrigon points QG-P13, QG-P14 and QG-P15 lie on QA-Co1.
  • The intersection point QG-P1.QG-P2 ^ QG-P12.QG-P14 ^ QL-P18.QL-P23 lies on QA-Co1.
  • Let P be a point on the Nine-point Conic in the Quadrigon-environment and let L be a line through P. Let L1, L2, L3, L4 be the lines of the Quadrigon where L1, L3 are opposite sides and L2, L4 are opposite sides.When P is the Midpoint of the line segment of L between L1 and L3 then automatically is P the Midpoint of the line segment of L between L2 and L4.
  • The 3 versions of QA-Co1 in a Quadrilateral have 3 common points being the vertices of triangle QL-Tr2.

 

 

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