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.
 
 
QA-P22-00
                      
Coordinates: 
1st CT-coordinate:
            (2 p + q + r) (q + r) (3 p2 + 3 p q + 3 p r - q r)
1st DT-coordinate:
            p2 (q2 + r2) - (q2 - r2)2
 
Properties:
  • QA-P22 lies on these QA-lines:
        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.
 

 

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