QL-P29: Circumcenter QL-X(265)-Quadrangle

 
QL-P29 is the Circumcenter of the circle through the four X(265) points of the Component Triangles of the Reference Quadrilateral, also called the QL-CT-versions of X(265).
X(265) = the Isogonal Conjugate of X(186), where:
X(186) is the Inverse of the Orthocenter X(4) in the circumcircle of a triangle.
See Ref-12 for an explanation of ETC-points X(i).
See Ref-33 Anopolis message # 409 for a discussion on this point.
There actually are 3 Triangle-points in ETC in the range X(1)-X(4000) for which their appearances in the Component Triangles of a Quadrilateral are concyclic: X(3), X(186), X(265). Only for X(4) these appearances are collinear in this range.
It’s remarkable that these points all relate to X(3) and X(4).
 
QL-P29 Circumcenter QL-X(265)-Quadrangle-01
Coordinates: 
1st CT-coordinate:
a4 l (-2 l m + m2 + 2 l n - n2)
+ (b2 - c2) (m - n) (b2 (l2 - 2 l m + m n) - c2 (l2 - 2 l n + m n))
- a2 c2 (l m2 + (l2 - 4 l m + m2) n + l n2)
+ a2 b2 (l m (l + m) - 4 l m n + (l + m) n2)
 
1st DT-coordinate:
a4 (m - n) (m + n) (-5 l4 + 3 m2 n2 + l2 (m2 + n2))
- b4 (m2 - n2)2 (l2 + 3 m2)
+ c4 (m2 - n2)2 (l2 + 3 n2)
+ 2 a2 b2 (-m4 n2 + 3 m2 n4 + l4 (m2 + n2) + l2 (3 m4 - 8 m2 n2 + n4))
- 2 a2 c2 ( 3 m4 n2 - m2 n4 + l4 (m2 + n2) + l2 (m4 - 8 m2 n2 + 3 n4))
 
Properties:
QL-P2.QL-P3              (-1 : 2 => QL-P29 = Reflection of QL-P3 in QL-P2)
QL-P4.QL-P20            (2 : -1 => QL-P29 = Reflection of QL-P4 in QL-P20)
QL-P22.QL-P27          (-3 : 4)

 

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