QA-P42: QA-Orthopole Center

 
QA-Tf3 is the QA-Orthopole Transformation that transforms a random point Q0 into another point Q1.
This point Q1 can be transformed by QA-Tf3 into another point Q2, etc..
The lines Qi.Qi+2 and Qi+1.Qi+3 intersect in a fixed point independent of starting point Q1.
Stated in another way: the consecutive generation points Qi diverge oscillating from a fixed point.
This point will be called QA-P42.
 
 
 
QA-P42-OrthopoleCenter-10 
 
 
 
Another interesting and related property is the similarity of Triangles of QA-points and their QA-Tf3-image:
 
 
QA-P42-OrthopoleCenter-20-SimilarTriangles
 
 
 
There is also a second major construction for this point.
The 4 centers of the circles through the pedal points of Quadrangle point Pi wrt Triangle Pj.Pk.Pl where (i,j,k,l) (1,2,3,4) form a QA-Pedal Quadrangle.
QA-P42 is the Homothetic Center of the 2nd generation QA-Pedal Quadrangle and the Reference Quadrangle.
The homothecy coefficient equals (2 cos α)2, where α is the angle formed by the asymptotes of QA-Co1 (this holds for convex quadrangles; a similar formula holds for the non-convex case). See Ref-36, pages 348,349.
 
QA-P42-OrthopoleCenter-01 
Coordinates:   
1st CT-Coordinate:
a2 q r (a2 q r + 3 c2 p q + 3 b2 p r) + 2 p2 (b2 SC r2 + c2 SB q2 + 2 S2 q r)
 
1st DT-Coordinate:
a4 (p2 - q2) (p2 - r2) + b2 p2 (p2 - q2) (b2 - 2 a2) + c2 p2 (p2 - r2) (c2 - 2 a2) + b2 c2 p2 (p2 + q2 + r2)
 
Properties:
        QA-P2.QA-P7                          (m : 1)
        QA-P4.QA-P8                          (m : 1)
        QA-P3.QA-Tf3(QA-P1)          ( n : 1)
        QA-P12.QA-Tf3(QA-P11)      ( n : 1)
        QG-P5.QA-Tf3(QA-P7)         ( n : 1)
        QL-P26.QG-P12
where (in CT-coordinates):
        m = (2 (c2 p q + b2 p r + a2 q r)2 - 8 p q r S2 (p + q + r))/ (    (c2 p q + b2 p r + a2 q r)2 + 12 p q r S2 (p + q + r))
          n  = (   (c2 p q + b2 p r + a2 q r)2 - 4 p q r S2 (p + q + r))/ (-4 (c2 p q + b2 p r + a2 q r)2)
Triangle QA-P2.QA-P42.QA-P11 is similar to Triangle QA-P4.QA-P42.QA-P12.
Triangle QA-P7.QA-P42.QA-P11 is similar to Triangle QA-P8.QA-P42.QA-P12.
 

 

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