nL-n-P8: nL-MVP Centroid

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nL-n-P8 is the nL-Mean Vector Point (see nL-n-Luc4) of X(2), the Triangle Centroid.

Another construction of nL-MVP Centroid:
The 4L-MVP-Centroid QL-P12 is constructed from 3L-MVP-Centroid X(2) using 4-polar Centroids.
The 5L-MVP-Centroid can be generated from 4L-MVP-Centroid QL-P12 in a similar way using 5-polar centroids.
Each line of the 5 lines in a 5-Line has 4 intersection points with the 4 other lines.
These 4 collinear points can be seen as a “flat” quadrangle and have a corresponding Quadrangle Centroid also called here a 4-polar Centroid.

In a 5-Line there are 5 Component Quadrilaterals. The lines connecting the QL-Centroids of these Component Quadrilaterals with their corresponding 4-polar centroids concur in one point being 5L-n-P8.

In the same way the 6L-MVP-Centroid also can be generated from the 5L-MVP-Centroid, etc.

Correspondence with ETC/EQF:

In a 3-Line:
3L-n-P8 = 3L-MVP Centroid = X(2)
3L-n-P9 = 3L-MVP Circumcenter = X(3)
3L-n-P10 = 3L-MVP Orthocenter = X(4)
3L-n-P11 = 3L-MVP Nine-point center = X(5)
In a 4-Line we find:
4L-n-P8 = 4L-MVP Centroid = QL-P12 (4L-Centroid)
4L-n-P9 = 4L-MVP Circumcenter = QL-P6 (Dimidium Point)
4L-n-P10 = 4L-MVP Orthocenter = QL-P2 (Morley Point)
4L-n-P11 = 4L-MVP Nine-point center = Midpoint (QL-P2,QL-P6)

Properties:

• nL-n-P8, nL-n-P9, nL-n-P10 and nL-n-P11 are collinear. Their mutual distance ratios correspond with the mutual distance ratios from triangle centers X(2), X(3), X(4) and X(5).
• nL-n-P8 is also the Homothetic Center of the Reference n-Line and the n-Line formed by the lines through the n (n-1)L-versions of nL-n-P8 parallel to the omitted Line. So starting with X(2) in a triangle it can be gradually constructed up to all higher n-levels in this way.

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