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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 [12] for an explanation of ETC-points X(i).

See [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).

Infovisual QL-P29-infovisual-cvt-01.png
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-Tf1(QL-P4) = The Reflection of QL-P1 in the Steiner line (QL-L2) and lies on the QL-X(265) circle.
  • Radius of the QL-X(265) circle = length line segment QL-P1.QP-P3.
  • Radius QL-X(186)circle : Radius QL-X(265)circle = QL-P1.QL-P28 : QL-P1.QL-P29.
  • The 5 versions of the QL-X(265) circle in a 5-Line (Pentalateral) all pass through a common point. See [34], QFG#82 by Seiichi Kirikami.
  • QL-P29 is the 4L-p2 point as described in Morley’s document [49], paragraph 3.
  • The QL-P29-Triple Triangle in a Quadrangle is perspective with the QL-P2-Triple Triangle as well as the QL-P3-Triple Triangle with perspector QA-P15.


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