QA-Cu6: QA-P1-Involution Center Cubic

This cubic is the locus of the involution centers of all lines passing through QA-P1 (Quadrangle-Centroid). So it is the locus of Midpoints of the Involutary Conjugates on the cubic QA-Cu5.

The involution is created by the intersection points of the QA-P1-line with the sides of Quadrigon P1.P2.P3.P4 (see chapter QA-Tf1: QA-Line Involution)

QA-P22, the Reflection of the Midpoint (QA-P1,QA-P5) in QA-P1 also lies on the cubic.

Similar cubics can be constructed by taking involution centers of lines passing through another point than QA-P1. The other 2 QA-quadrigons deliver the same cubic. This makes the cubic a real QA-Cubic.

Equation CT-notation:

r (p+q+2r) ((3q+p)x–(3p+q) y) x y

+ q (p+2q+r) ((3p+r) z–(3r+p) x) z x

+ p (2p+q+r) ((3r+q) y–(3q+r) z) y z – (p-q)(q-r)(r-p) x y z = 0

Equation DT-notation:

q² r² (q² – r²) x³ + q² r² (2 p² (y – z) – (q² – r²) (y + z)) x²

+ p² r² (r² – p²) y³ + p² r² (2 q² (z – x) – (r² – p²) (x + z)) y²

+ p² q² (p² – q²) z³ + p² q² (2 r² (x – y) – (p² – q²) (x + y)) z² = 0

Properties

• The vertices of the Quadrangle and the Midpoints of the Diagonal triangle and QA-P1, QA-P22 lie on this cubic.
• The intersection point QA-P1.QA-P6 ^ QA-P22.QA-P29 lies on this cubic.
• The intersection points of the lines through QA-P1 parallel to the QA-sidelines with the opposite sidelines all lie on this cubic.

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