nP-n-Luc1 nP-Mean Vector Point

A Mean Vector Point (MVP) is the mean of a bunch of n vectors with identical origin.
It is constructed by adding these vectors and then dividing the Sumvector by n.
The Mean Vector Point is the endpoint of the divided Sumvector.
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-Line (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.

Preservation of 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-Lines, usually collinearity of MVP-points will not be preserved. The mentioned triangle centers on the Eulerline are exceptions.