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.
+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)
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)
- QL-P2 lies on these lines:
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 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.