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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
Additional comment
  • 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:


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