QA-P22

QA-P22: Midpoint QA-P1 and QA-P20

QA-P22 is the Midpoint of QA-P1 (QA-Centroid) and QA-P20 (Reflection of QA-P5 in QA-P1).

It is also the Center of the Involution created by the intersection of QA-P1.QA-P5 with the 6 Quadrangle lines (see notes below).

Construction:

Construct the complement of QA-P1 wrt the QA-Diagonal Triangle.

Coordinates:

1^st CT-coordinate:

(2 p + q + r) (q + r) (3 p² + 3 p q + 3 p r - q r)

1^st DT-coordinate:

p² (q² + r²) - (q² - r²)²

Properties:

– QA-P1.QA-P5 (-1 : 3)

– QA-P3.QA-P35 ( 5 : -1)

QA-P22 is the complement of QA - P1 wrt the QA-Diagonal Triangle..
QA-P22 is the Center of the Involution on the line QA-L3 = QA-P1.QA-P5. This Line Involution is defined by 2 pairs of points being the intersection points of L with the opposite sides of a component quadrigon of the Reference Quadrangle. The result is the same for all 3 component quadrigons of the Reference Quadrangle. QA-P1 and QA-P20 are the two double points of this Line Involution.
QA-P22 lies on the cubic QA-Cu6 (QA-P1-InvolutionCenter Cubic).
QA-P22 is the QA-Centroid of the Quadrangle formed by the vertices of the QA-Diagonal Triangle and QA-P20 (Reflection of QA-P5 in QA-P1).
QA-P22 = QA-Centroid of the DT-vertices + QA-P20.
QA-P22 = QA-Centroid of the Quadrangle QA-P2.QA-P12.QA-P20.QA-P30.

MATHEMATICS