QL-P2: Morley Point

 
Let Ni be Nine-point Center of triangle LjLkLl.
Let Lpi be the perpendicular line of Ni at Li.
Now all perpendicular lines Lpi (i=1,2,3,4) concur in one point QL-P2.
This point was found by Frank Morley naming it the Second Orthocenter in his document Ref-49, paragraph 3. It is described as a recursive point in an n-Line. In his document he uses the letter “h” for this point.
QL-P2 is also mentioned by J.W. Clawson in Ref-31 (pp. 40 and 41) as the “mean center of gravity of equal masses placed at H1, H2, H3, H4”, where H1, H2, H3, H4 are the orthocenters of the Component Triangles of the Reference Quadrilateral.
 
QL-P2-Morley-Point-00
Coordinates: 
1st CT-coordinate:
            +a4   l   (2 l - m - n) + b2 c2 (2 l2 - 3 l m - 3 l n + 4 m n)
- b4 (l - 2 m) (l - n) + a2 c2 (l2 m + l2 n - 5 l m n + 2 m2 n + l n2) /(m - n)
- c4 (l - 2 n) (l - m) + a2 b2 (l2 m + l2 n - 5 l m n + 2 m n2 + l m2)/(n - m)
 
1st DT-coordinate:
            Sb Sc - (Sb2 m2 (-l2+n2))/((l2-m2) (m2-n2)) - (Sa Sb m2)/(l2-m2)
         - (Sc2 n2 (l2-m2))/((l2- n2 ) (m2-n2)) - (Sa Sc n2 )/(l2- n2)
 
 
Properties:
  • QL-P2 lies on these lines:
        QL-P4.QL-P22    (2 : -1 => QL-P2 = Reflection of QL-P4 in QL-P22)
        QL-P6.QL-P12     (3 : -2)
        QL-P7.QL-P9
        QA-P24.QG-P5    (1 : 1 => QL-P2 = Midpoint of QA-P24 in QG-P5)
  • The QA-Triangle formed by the 3 QA-versions of QL-P2 is Triangle QA-Tr3 with Centroid QA-P14 and Orthocenter QA-P15.
  • QL-P2 is the Centroid of the Triangle formed by the 3 QL-versions of QA-P14 (note Eckart Schmidt).
  • QL-P2 is the Centroid of the Orthocenters of the 4 Component Triangles of the Reference Quadrilateral (Eckart Schmidt, September 18, 2012).
  • QL-P2 is the Orthocenter of QL-Tr2, the triangle formed by the three common points of the 3 QL-versions of the Nine Point Conic QA-Co1.
  • QL-P2 lies on the Circle defined by the 3 QL-versions of QG-P7 (note Eckart Schmidt).
  • The Clawson-Schmidt Conjugates of QL-P2 lies on QL-Ci3 (Miquel Circle).
  • QL-P2 (Morley Point) lies on the circumcircle of the triangle formed by the 1st Quasi Nine-point Centers (QG-P7) of the 3 QL-Quadrigons.
  • QL-P1 lies on the Polar (see Ref-13, Polar) of QL-P2 wrt QL-Co1.
  • QL-P2 is the External Homothetic Center of Morley’s Second Circle (see QL-P30) and the Miquel Circle QL-Ci3. See Ref-49, Theorem 10.
  • The QL-P2 Triple Triangle in a Quadrangle is Orthologic wrt all QA-Component Triangles (Seiichi Kirikami, Ref-34, QFG #980, # 982). See QA-Tr-1.
  • The QL-P2-Triple Triangle in a Quadrangle is perspective with the QL-P3-Triple Triangle as well as the QL-P29-Triple Triangle in a Quadrangle with perspector QA-P15.

 
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