nL-n-Ci1: nL-Center Circle
A Triangle (3-Line) has a circumcircle. Morley in Ref-49 calls this circle a Centercircle.
In a Quadrilateral (4-Line) there are 4 component 3-Lines whose 3L-Centercircle Centers are concyclic on the 4L-Centercircle.
In a Pentalateral (5-Line) there are 5 component 4-Lines whose 4L-Centercircle Centers are concyclic on the 5L-Centercircle. Etc.
Morley proved in Ref-49 that there exists a Centercircle in an n-Line for all n, built from the centers of the Centercircles from the Component m-Lines.
The Center of this Centercircle is nL-n-P3.
Correspondence with ETC/EQF:
When n=3, then nL-n-Ci1 = Triangle Circumcircle.
When n=4, then nL-n-Ci1 = Miquel Circle QL-Ci3.
When n=5, then nL-n-Ci1 = Circumcircle of the concyclic five versions of QL-P4.
Properties:
• Each Oi.Oj-intercepted inscribed nL-n-Ci1-angle = Angle(Li,Lj) mod ,
where (i,j) are different numbers from (1, … ,n).
Note that intercepted inscribed angles in a circle are twofold: and - .
When taken mod the angles are and -. Anyway the circle is the locus of points which form inscribed angles (mod ) with a line segment, + when occurring on one side of the line segment and - when occurring on the other side.
The same is true for angles between two intersecting lines. They are twofold and when taken mod they are + and -.
Example: let V be variable point on nL-n-Ci1, now the twofold angle Oi.V.Oj = twofold angle (Li,Lj). See Ref-34, QFG#1893.
• When n=5 (in a 5-Line) 5L-s-P2 lies on the Centercircle 5L-n-Ci1.
• Each Oi.Oj-intercepted inscribed nL-n-Ci1-angle = Angle(Li,Lj) mod ,
where (i,j) are different numbers from (1, … ,n).
Note that intercepted inscribed angles in a circle are twofold: and - .
When taken mod the angles are and -. Anyway the circle is the locus of points which form inscribed angles (mod ) with a line segment, + when occurring on one side of the line segment and - when occurring on the other side.
The same is true for angles between two intersecting lines. They are twofold and when taken mod they are + and -.
Example: let V be variable point on nL-n-Ci1, now the twofold angle Oi.V.Oj = twofold angle (Li,Lj). See Ref-34, QFG#1893.
• When n=5 (in a 5-Line) 5L-s-P2 lies on the Centercircle 5L-n-Ci1.