QL-P28: Circumcenter QL-X(186)-Quadrangle
QL-P28 is the Circumcenter of the circle through the four X(186) points of the Component Triangles of the Reference Quadrilateral also called the QL-CT-versions of X(186).
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).
-Quadrangle-01.png)
Coordinates:
1st CT-coordinate:
a2 (a6 (l - m) (l - n) (2 l - m - n) (m - n)
+ b6 (l3 m - 3 l2 m2 - m n (3 m2 - 3 m n + n2) + l (2 m3 + 3 m2 n - 3 m n2 + n3))
- c6 (l3 n - 3 l2 n2 - m n (m2 - 3 m n + 3 n2) + l (m3 - 3 m2 n + 3 m n2 + 2 n3))
+ b2 c4 ( 9 l m n2 - 3 l2 n (2 m + n) + l3 (m + 2 n) + m n (m2 - 3 m n - n2))
+ b4 c2 (-9 l m2 n - l3 (2 m + n) + 3 l2 m (m + 2 n) + m n (m2 + 3 m n - n2))
+ 3 a2 b4 (l - m) (l - n) (m - n) (m - n)
- 3 a2 c4 (l - m) (l - n) (m - n) (m - n)
+ a4 b2 (l3 (-3 m + 2 n) + m n (m2 + 3 m n - 3 n2) + 3 l2 (m2 + 2 m n - 2 n2) + 3 l n (-3 m2 + m n + n2))
- a4 c2 (l3 (2 m - 3 n) + 3 l m (m2 + m n - 3 n2) + m n (-3 m2 + 3 m n + n2) + l2 (-6 m2 + 6 m n + 3 n2))
+ a2 b2 c2 (m - n) (l3 + 9 l m n - 3 l2 (m + n) - 2 m n (m + n)))
1st DT-coordinate:
+ a8 (l2 - m2) (l2 - n2) (m2 - n2) (5 l4 - l2 m2 - l2 n2 - 3 m2 n2)
+ b8 (l2 - m2) (l2 + 3 m2) (m - n)3 (m + n)3
+ c8 (l2 - n2) (l2 + 3 n2) (m - n)3 (m + n)3
- 4 b6 c2 l2 (l2 - m2) (m - n)3 (m + n)3
- 4 b2 c6 l2 (l2 - n2) (m - n)3 (m + n)3
+ 6 b4 c4 (l2 - m2) (l2 - n2) (m - n)3 (m + n)3
+ 6 a4 b4 (l2 - m2) (l2 - n2) (m2 - n2) (3 l2 m2 + m4 - l2 n2 - 3 m2 n2)
- 6 a4 c4 (l2 - m2) (l2 - n2) (m2 - n2) (l2 m2 - 3 l2 n2 + 3 m2 n2 - n4)
+ 4 a2 b6 (l2 - m2) (-4 l2 m6 + 4 l4 m2 n2 + l2 m4 n2 + 7 m6 n2 - 6 l2 m2 n4 - 6 m4 n4 + l2 n6 + 3 m2 n6)
- 4 a2 c6 (l2 - n2) (l2 m6 + 4 l4 m2 n2 - 6 l2 m4 n2 + 3 m6 n2 + l2 m2 n4 - 6 m4 n4 - 4 l2 n6 + 7 m2 n6)
+ 4 a6 c2 (l2 - n2) (2 l6 m2 +3 l4 m4 - l2 m6 + 2 l6 n2 - 13l4 m2 n2 + 6 l2 m4 n2 - 3 m6 n2 + 2 l4 n4 - l2 m2 n4 + 3 m4 n4)
- 4 a6 b2 (l2 - m2) (2 l6 m2 + 2 l4 m4 + 2 l6 n2 -13 l4 m2 n2 - l2 m4 n2 + 3 l4 n4 + 6 l2 m2 n4 + 3 m4 n4 - l2 n6 - 3 m2 n6)
- 4 a2 b4 c2 (-l4 m6 - 3 l2 m8 + 12 l6 m2 n2 - 18 l4 m4 n2 + 15 l2 m6 n2 + 3 m8 n2 - 21 l4 m2 n4 + 27 l2 m4 n4 - 18 m6 n4 + 4 l4 n6 - 3 l2 m2 n6 + 3 m4 n6)
+ 4 a2 b2 c4 (4 l4 m6 + 12 l6 m2 n2 - 21 l4 m4 n2 - 3 l2 m6 n2 - 18 l4 m2 n4 + 27 l2 m4 n4 + 3 m6 n4 - l4 n6 + 15 l2 m2 n6 - 18 m4 n6 - 3 l2 n8 + 3 m2 n8)
- 4 a4 b2 c2 (m2 - n2) (3 l6 m2 + 5 l4 m4 + 3 l6 n2 - 16 l4 m2 n2 - 3 l2 m4 n2 + 5 l4 n4 - 3 l2 m2 n4 + 6 m4 n4)
Properties:
- The QL-X(186) circle is the Clawson-Schmidt Conjugate (QL-Tf1) of the QL-X(265) circle.
- QL-Tf1(QL-P28) = Inverse of QL-P1 in QL-X(186) circle. QL-Tf1(QL-P29) = Inverse of QL-P1 in QL-X(265) circle.
- QL-P29 lies on the line QL-P1.QL-Tf1(QL-P28), as well as QL-P28 lies on the line QL-P1.QL-Tf1(QL-P29),
- Moreover: QL-P28.QL-Tf1(QL-P29) : QL-Tf1(QL-P29).QL-P1 = QL-P29.QL-Tf1(QL-P28) : QL-Tf1(QL-P28).QL-P1.
- QL-P28.QL-P4 // QL-P1.QL-Tf1(QL-P3)
- QL-P4 lies on the QL-X(186)-circle.
- 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(186) circle in a 5-Line (Pentalateral) all pass through a common point. See Ref-34, QFG#82 by Seiichi Kirikami.