nL-n-gi: nL-Morley's intermediate recursive gi points

---

nL-n-gi is defined as the Centroid of n points (n-1)L-n-p(i-1).
Note that the letter “p” is in lower case. In Morley’s document Ref-49 it is denoted as “pi”.
Morley uses it as intermediate point(s) to make it possible to construct his so called first Orthocenter (nL-o-P1) as well as his so called Ortho Directrix (nL-e-L1).
It is used in the definition of nL-n-p1.
nL-n-pi = Ratiopoint of
          nL-n-p(i-1) and
          nL-n-g(i-1) being the Centroid of n points (n-1)L-n-p(i-1)
          with ratio n : (i-n).

Serial steps of construction
The meaning of Morley’s intermediate recursive pi- and gi-points can best be understood in writing down the first serial steps for increasing n.

In a 3-Line:
          The Circumcenter of the 3 vertices is 3L-n-p0. = 3L-n-P3
          The Centroid of the 3 points 2L-n-p0 is 3L-n-g0.
          The Ratiopoint 3L-n-p0.3L-n-g0 (3:-2) is 3L-n-p1. = 3L-n-P7 = 3L-o-P1
In a 4-Line:
          The Circumcenter of the 4 points 3L-n-p0 is 4L-n-p0. = 4L-n-P3
          The Centroid of the 4 points 3L-n-p0 is 4L-n-g0.
          The Ratiopoint 4L-n-p0.4L-n-g0 (4:-3) is 4L-n-p1. = 4L-n-P7 = 4L-e-P1
In a 5-Line:
          The Circumcenter of the 5 points 4L-n-p0 is 5L-n-p0. = 5L-n-P3
          The Centroid of the 5 points 4L-n-p0 is 5L-n-g0.
          The Ratiopoint 5L-n-p0.5L-n-g0 (5:-4) is 5L-n-p1. = 5L-n-P7
          The Centroid of the 5 points 4L-n-p1 is 5L-n-g1.
          The Ratiopoint 5L-n-p1.5L-n-g1 (5:-3) is 5L-n-p2. = 5L-o-P1
In a 6-Line:
          The Circumcenter of the 6 points 5L-n-p0 is 6L-n-p0. = 6L-n-P3
          The Centroid of the 6 points 5L-n-p0 is 6L-n-g0.
          The Ratiopoint 6L-n-p0.6L-n-g0 (6:-5) is 6L-n-p1. = 6L-n-P7
          The Centroid of the 6 points 5L-n-p1 is 6L-n-g1.
          The Ratiopoint 6L-n-p1.6L-n-g1 (6:-4) is 6L-n-p2. = 6L-e-P1
In a 7-Line:
          The Circumcenter of the 7 points 6L-n-p0 is 7L-n-p0. = 7L-n-P3
          The Centroid of the 7 points 6L-n-p0 is 7L-n-g0.
          The Ratiopoint 7L-n-p0.7L-n-g0 (7:-6) is 7L-n-p1. = 7L-n-P7
          The Centroid of the 7 points 6L-n-p1 is 7L-n-g1.
          The Ratiopoint 7L-n-p1.7L-n-g1 (7:-5) is 7L-n-p2.
          The Centroid of the 7 points 6L-n-p2 is 7L-n-g2.
          The Ratiopoint 7L-n-p2.7L-n-g2 (7:-4) is 7L-n-p3. = 7L-o-P1
In a 8-Line:
          The Circumcenter of the 8 points 7L-n-p0 is 8L-n-p0. = 8L-n-P3
          The Centroid of the 8 points 7L-n-p0 is 8L-n-g0.
          The Ratiopoint 8L-n-p0.8L-n-g0 (8:-7) is 8L-n-p1. = 8L-n-P7
          The Centroid of the 8 points 7L-n-p1 is 8L-n-g1.
          The Ratiopoint 8L-n-p1.8L-n-g1 (8:-6) is 8L-n-p2.
          The Centroid of the 8 points 7L-n-p2 is 8L-n-g2.
          The Ratiopoint 8L-n-p2.8L-n-g2 (8:-5) is 8L-n-p3. = 8L-e-P1

As can be seen always nL-n-p0 = nL-n-P3 and nL-n-p1 = nL-n-P7.
For even n, nL-n-p((n/2)-1) = nL-e-P1.
For odd n, nL-n-p((n-1)/2) = nL-o-P1.
After all the whole circus with pi- and gi-points is developed by Morley for constructing nL-o-P1 and nL-e-P1. A bycatch is that nL-n-p1 = nL-n-P7, but nL-n-P7 also can be constructed as a vectorsum (see nL-n-Luc3 and nL-n-P7).

See also the notes at nL-n-pi.

Correspondence with ETC/EQF:
• In a 3-Line:
          3L-n-g0 = 3L-n-g1 = 3L-n-g2 = 3L-n-g3 = X(2).
• In a 4-Line:
          4L-n-g0 = QL-P6
          4L-n-g1 = QL-P2
          4L-n-g2 = QL-P6
          4L-n-g3 = QL-P12

---