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.

Construction of Axes/Asymptotes of QA-Co1:

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

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)) )

Equation DT-notation:

r² x y + q² x z + p² y z = 0

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. For the construction of the Involutary Conjugate of some Infinity Point see QA-Tf2.
• 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)
  • 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 [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.
• QA-P16 is the perspector the QA-Diagonal Triangle (QA-Tr1) and the tangential triangle of QA-Co1 wrt the QA-Diagonal Triangle (note Randy Hutson).
• The 3 versions of QA-Co1 in a Quadrilateral have 3 common points being the vertices of triangle QL-Tr2.

Estimated human page views: 1037