QA-L1: QA-P1-P2-P3 Line

The QA-L1-line is the line through QA-P2 (Euler-Poncelet Point) and QA-P3 (Gergonne-Steiner Point). Both points are constructed in a similar way. They are both common points of circles through midpoints of line segments between Quadrangle points.
QA-P1 (Centroid) also lies on this line and is Midpoint (QA-P2,QA-P3).
Of next 4 points on QA-L1 it also appears that:
• QA-P1 is the Euler-Poncelet Point of the Nine-point Center Quadrangle.
• QA-P2 is the Euler-Poncelet Point of the Orthocenter Quadrangle.
• QA-P3 is the Euler-Poncelet Point of the Circumcenter Quadrangle.
• QA-P34 is the Euler-Poncelet Point of the Centroid Quadrangle.
QA-P1, QA-P2, QA-P3 and QA-P34 have mutual distance ratios similar to their corresponding points in the Triangle Environment on the Euler Line. Coefficients:
1st CT-coefficient:
a4 q (q - r)r/(q+r)  +  b4 p r (p+2q+r)/(p+r)  –  c4 p q(p+q+2r)/(p+q)
–  b2 c2 p (q-r)        +  a2 c2 q (p+3r)                  – a2 b2 r (p+3q)

1st DT-coefficient:
p2 (b2 r2 - c2 q2)  (b2 r2 (p2 + q2 - r2) + c2 q2 (p2 - q2 + r2) - 2 a2 q2 r2)

Properties:
• QA-L1 is the QA-L2 line of the Nine-point Center Quadrangle (Eckart Schmidt, August 24, 2012).
• The Orthopole (see Ref-13) of QA-P3.Pi wrt triangle Pj.Pk.Pl, for all (i,j,k,l) ∈ (1,2,3,4) lie on the circle with diameter QA-L1-line segment QA-P2.QA-P3. See Ref-11, Hyacinthos messages 21865 & 21867.

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