nL-n-Luc4 nL-Mean Vector Point

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 nL-n-P8 to nL-n-P11.

Resemblance with nL-n-Luc3
nL-n-Luc4 looks like nL-n-Luc3. In both cases, a Sum vector is used. Only in nL-n-Luc4 the Sum vector is divided by the number of vectors.

Origin independent
It is most special that with the definition of nL-n-Luc4 the location of the origin is unimportant.
In all n-Lines 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-Line by a simple recursive construction. The resulting point which will be called an nL-MVP Center, where MVP is the abbreviation for Mean Vector Point.
When X(i) is a triangle Center we define the nL-MVP X(i)-Center as the Mean Vector Point of the n (n-1)L-MVP X(i)-Centers.
When the (n-1)L-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 3L-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 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 nL-n-Luc4(X(i))
An nL-Mean Vector Point of some Triangle Center X(i) also can be constructed as the Centroid of the corresponding (n-1)L-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 3L-MVP X(i)-center, which simply is the X(i) Triangle Center.

Preservation of distance ratios
Collinearity of points is not preserved by nL-n-Luc4. However, the exception is that when collinear points have fixed distance ratios, then after transformation by nL-n-Luc4 collinearity of points including their mutual distance ratios will be preserved.
See also notes on this similar subject at nP-n-Luc1 and Ref-34, QFG#3499, #3500.
Therefore the Centroid, Circumcenter, Orthocenter, and Nine-point Center are when transferred to an n-Line collinear and their mutual distance ratios will be preserved.
This is deviating from Morley’s Centroid, Circumcenter, Orthocenter and Nine-point Center (resp. nL-n-P2, nL-n-P3, nL-n-P4, nL-n-P5) in an n-Line. Clearly, they are collinear, but their mutual distance ratios in this special case are not preserved. See nL-n-P2.