QL-P17: QL-Adjunct Quasi Circumcenter


QL-P17 is the 2nd common intersection point of the coaxal circumcircles of the 3 triangles formed by the 3 QL-versions of resp. QG-P1, QG-P5, QG-P9.
The 1st intersection point of these circles is QL-P16 (QL-Quasi Circumcenter).

QL-P17-Adjunct-Quasi Circumcenter-01

Coordinates:
1st CT-coordinate:
            m n (a2 ( l - m) ( l - n) (2 l2 (m - n)2 - m2 n2 + l m n (m + n))
                  + b2 (m - n) (m - l) n l (-3 l m + l n + m n)
                  + c2 (n - m) (n - l) m l (l m - 3 l n + m n)))

1st DT-coordinate:
            a2 m2 n2 / (m2-n2)

Properties:
        QL-P8.QL-P25                       (-2 : 3)
        QL-P13.QL-P24
        QA-P3.QG-P13
  • QL-P17 is the AntiComplement of QL-P25 wrt the QL-Diagonal Triangle.
  • QL-P25 lies on the polar of QL-P17 wrt the Polar Circle of the QL-Diagonal Triangle (note Eckart Schmidt).
  • QL-P17 lies on the circumcircles of triangles formed by the 3 QL-versions of QG-P1 (being QL-Diagonal Triangle QL-Ci1), QG-P5 as well as QG-P9.
  • QL-P17 lies on the Dimidium Circle QL-Ci6. The second intersection point with the circumcircle of the QL-Diagonal Triangle (QL-Ci1) is QL-P24.
  • QL-P17 is the Miquel Point (QL-P1) of the Quadrilateral formed by the lines of the QL-Diagonal Triangle and the Newton Line (QL-L1). See Ref-34, message # 164 and accompanying document “Diagonal Quadrilaterals and Conics through the points S1,S2,S3” page 2 of Bernard Keizer.