nL-n-P1: nL-Centric Focus

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. Moreover the 4 3L-Centercircles have a common point, the 4L-Centric Focus.

In a Pentalateral (5-Line) there are 5 Component 4-Lines whose 4L-Centercircle centers are concyclic on the 5L-Centercircle. Again the 5 4L-Centercircles have a common point, the 5L-Centric Focus.

Etc.

Goormaghtigh (Ref-55) named this point the Centric Focus because in a 4-Line this point is the Focus of the inscribed parabola (QL-Co1). In a 4-Line it also is the node of Morley’s Mono Cardioid (QL-Qu1).

Moreover nL-n-P1 is the node of the generalized Mono Cardioid also called the nL-Mono EnnaCardioid described by Morley in Ref-47. This nL-Mono EnnaCardioid is a curve circumscribing all (n-1)L-Centercircles. See nL-n-Cv1.

Correspondence with ETC/EQF

• When n=4, then nL-n-P1 = QL-P1.

Properties

• 5L-n-P1 is collinear with 5L-o-P2 and 5L-n-P3.
• 5L-n-P1 is a point on 5L-o-Ci1 and inversion of 5L-o-P2 wrt 5L-n-Ci1 (QFG#722, October 6, 2014, Eckart Schmidt).
• 5L-n-P1 is a node of the Mono EnnaCardioid nL-n-Cv1 circumscribing n (n-1)L-EnnaCardioids (n-1)L-n-iCv1. See Ref-37.

Estimated human page views: 284