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.
Of next 4 points on QA-L1 it also appears that:
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.
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)
p2 (b2 r2 - c2 q2) (b2 r2 (p2 + q2 - r2) + c2 q2 (p2 - q2 + r2) - 2 a2 q2 r2)
- 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.