A Mean Vector Point (MVP) is the mean of a bunch of n vectors with an identical origin.
It is constructed by adding these vectors and then dividing the Sum vector by n.
The Mean Vector Point is the endpoint of the divided Sum vector.
This method is used for nP-n-P1 to nP-n-P4.
It is constructed by adding these vectors and then dividing the Sum vector by n.
The Mean Vector Point is the endpoint of the divided Sum vector.
This method is used for nP-n-P1 to nP-n-P4.
Origin independent
It is most special that with the definition of nP-n-Luc1 the location of the origin is unimportant.
In all n-Points we can use any random point as origin. The endpoint of the resultant vector will be the same for all different origins.
Recursive application
Every Triangle Center can be transferred to a corresponding point in an n-Point by a simple recursive construction. The resulting point which will be called an nP-MVP Center, where MVP is the abbreviation for Mean Vector Point.
When X(i) is a triangle Center we define the nP-MVP X(i)-Center as the Mean Vector Point of the n (n-1)P-MVP X(i)-Centers.
When the (n-1)P-MVP X(i)-Centers aren’t known they can be constructed from the MVP X(i)-Centers another level lower, according to the same definition. By applying this definition to an increasingly lower level finally the level is reached of the 3P-MVP X(i)-Center, which simply is the X(i) Triangle Center.
See Ref-34, QFG#869,#873,#878,#881.
Universal Level-up construction
Unlike other Level-up constructions, this construction probably can be applied to all Central Points at all levels.
Consequently, all known ETC-points and all known EQF-points will have a related MVP-point in every n-Point (n>3,4).
Another general construction of nP-n-Luc1(X(i))
An nP-Mean Vector Point of some Triangle Center X(i) also can be constructed as the Centroid of the corresponding (n-1)P-Mean Vector Points of some Triangle Center X(i). Again by applying this definition to an increasingly lower level finally the level is reached of the 3P-MVP X(i)-center, which simply is the X(i) Triangle Center.
An nP-Mean Vector Point of some Triangle Center X(i) also can be constructed as the Centroid of the corresponding (n-1)P-Mean Vector Points of some Triangle Center X(i). Again by applying this definition to an increasingly lower level finally the level is reached of the 3P-MVP X(i)-center, which simply is the X(i) Triangle Center.
Preservation of fixed distance ratios
The Centroid, Circumcenter, Orthocenter, and Nine-point Center are when transferred to an n-Point collinear and their mutual distance ratios are preserved.
However when Triangle Centers (other than X(2), X(3), X(4), X(5)) are transferred to higher level n-Points, usually collinearity of MVP-points will not be preserved. The mentioned triangle centers on the Eulerline appear to be exceptions.
At Ref-34, QFG#3499 Stanley Rabinowitz noted that in a 4P configuration the Triangle Centers derived from I=X(1), M=X(2), N=X(8) and S=X(10) in a triangle lying on the Nagel Line with fixed distances IM : MS : SN = 2 : 1 : 3, the distances for the corresponding 4P-points are preserved as well.
At Ref-34, QFG#3500 Vu Thanh Tung comments:
“For each i = 1 : k, let Ui, Vi, Wi be three collinear points on the plane such that UiVi : ViWi = t = const.
Let U, V, W be the kP-n-P1 (centroid) of k points (Ui), (Vi), (Wi) respectively then U, V, W are collinear and UV : VW = t.”
This is actually a description of nP-n-Luc1 as well as a concise proof that nP-n-Luc1 preserves fixed distance ratios on a line.
Summarizing:
Collinearity of points is not preserved by nP-n-Luc1. However, the exception is that when collinear points have fixed distance ratios, then after transformation by nP-n-Luc1 collinearity of points including their mutual distance ratios will be preserved.
At Ref-34, QFG#3500 Vu Thanh Tung comments:
“For each i = 1 : k, let Ui, Vi, Wi be three collinear points on the plane such that UiVi : ViWi = t = const.
Let U, V, W be the kP-n-P1 (centroid) of k points (Ui), (Vi), (Wi) respectively then U, V, W are collinear and UV : VW = t.”
This is actually a description of nP-n-Luc1 as well as a concise proof that nP-n-Luc1 preserves fixed distance ratios on a line.
Summarizing:
Collinearity of points is not preserved by nP-n-Luc1. However, the exception is that when collinear points have fixed distance ratios, then after transformation by nP-n-Luc1 collinearity of points including their mutual distance ratios will be preserved.