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.

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 b² r (p+3q) + a² c² q (p+3r)

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 = QA-P1.QA-P2 // QA-P22.QA-P29
• QA-L1 is the QA-L2 line of the Nine-point Center Quadrangle (Eckart Schmidt, August 24, 2012).
• The Orthopole (see [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 [11] Hyacinthos messages 21865 & 21867.

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