nL-n-Luc5a   nL-Ref/Par1 construction

nL-n-Luc5a is a Level-up construction which uses the Reference n-Line Ref and the 1st generation parallel n-Line Par1 wrt some lower-level center to construct a Perspective / Homothetic Center. See nL-n-Luc5. Examples:

 3L-point 4L-point 5L-point 6L-point 3L-n-P8 = X(2) 4L-n-P8 (Ref/Par1-HC, Ref=Par2) 5L-n-P8 (Ref/Par1/Par2-HC) 6L-n-P8 (Ref/Par1/Par2-HC) Etc. 3L-n-P3/P9 = X(3) 4L-n-Px (Ref/Par1-HC, Ref=Par2) No Ref/Par1-relationship. No Ref/Par2-relationship. No Par1/Par2-relationship. 3L-n-P4/P10 = X(4) 4L-n-Px (Ref/Par1-HC, Ref=Par2) 5L-s-P7 (Par1/Par2-HC) No Ref/Par1-relationship. No Ref/Par2-relationship. No Ref/Par1-relationship. No Ref/Par2-relationship. No Par1/Par2-relationship. 3L-n-P5/P11 = X(5) 4L-n-P2 (Ref/Par1-HC, Ref=Par2) No Ref/Par1-relationship. No Ref/Par2-relationship. No Par1/Par2-relationship. Points on line 4L-n-P8.4L-n-P5 4L-n-P8 = QL-P12 4L-n-P5 = QL-P30 4L-n-P5. 4L-n-P8 (2:-1)= QL-P27 Points on line 5L-n-P8. 5L-n-P2 preserving distance ratio’s (Ref/Par1/Par2-HC) 5L-n-P8 5L-n-P2 5L-n-P2. 5L-n-P8 (2:-1) No Ref/Par1-relationship. No Ref/Par2-relationship. No Par1/Par2-relationship.

Note that the mutual distance ratios of points X(2), X(3), X(4), X(5) (lying on the 3L-Eulerline) and resp. of QL-P12, QL-P5, QL-P20, QL-P22 (lying on the 4L-Newton Line) are identical.

Summary

General cases
nL-n-Luc5a (nL-n-P8) = pL-n-P8, where p=(n+1)
nL-n-Luc5a (nL-n-P5) = pL-n-P2, where p=(n+1)
nL-n-Luc5a (nL-n-Px) = pL-n-Py, where p=(n+1) and
nL-n-Px = Ratiopoint nL-n-P8.nL-n-P5 (s:t)
pL-n-Py = Ratiopoint pL-n-P8.pL-n-P2 (s:t),
meaning that the line nL-n-P8.nL-n-P5 is leveled-up into the line pL-n-P8.pL-n-P2
and that distance ratios are preserved.
Specific cases
nL-n-Luc5a (X(3)) = QL-P5, there is no nL-n-Luc5a (QL-P5)
nL-n-Luc5a (X(4)) = QL-P20, there is no nL-n-Luc5a (QL-P20)
nL-n-Luc5a (X(5)) = QL-P22, there is no nL-n-Luc5a (QL-P22)