QA-P6: Parabola Axes Crosspoint

 
The Parabola Axes Crosspoint is the intersection point of the axes of the 2 parabolas that can be constructed through P1, P2, P3, P4.
It is also is the Midpoint of the Euler-Poncelet Point QA-P2 and the Isogonal Center QA-P4.
Because these parabolas only can be constructed when the Reference Quadrangle is not concave a better definition of this point is: “the Midpoint of the Euler-Poncelet Point and the Isogonal Center”.
Because the property related to the parabolas is much more appealing this point is named after its primary function.
It also can be reasoned that in a concave quadrangle this point represents the intersection point of the axes of the imaginary parabolas.
QA-P6-ParabolaAxisCrossPoint-00
 
Coordinates: 
1st CT-Coordinate:
a4 q2 r2 + c2 p2 q2 SB + b2 p2 r2 SC - p q r (a2 (p + q + r) SA + 2 (p S2 - q SB2 - r SC2))
1st DT-Coordinate:
                p2 (b2 SB r2 + c2 SC q2 - c2 b2 p2)
 
Properties:
  • QA-P6 lies on these QA-lines:
        QA-P1.QA-P23
        QA-P2.QA-P4              ( 1 : 1 => QA-P6 = Midpoint QA-P2.QA-P4)
        QA-P28.QA-P29         (-1 : 2 => QA-P6 = Reflection of QA-P29 in QA-P28)
  • QA-P6 is the Involutary Conjugate (see QA-Tf2) of QA-P30.
  • QA-P6 lies on the Simson Line (QA-P6.QA-P36) of QA-P2 occurring on the circumcircle of the QA-Diagonal Triangle.
  • QA-P6 is the common point of the 4 Simson Lines of QA-P2 wrt the medial triangles of the component triangles (see Ref-34, QFG#548-550).
  • For all QA-Quadrigons QA-P6 is the center of the Pedal Quadrangle of QA-P4 (Isogonal Center) , which is a parallelogram. See Ref-15 f, theorem (23).
  • The QA-Orthopole (QA-Tf3) of QA-P6 is the Midpoint(QA-P2,QA-P23).

 
 

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