2P-s-iRg1 Regular LH-n-Gons (Lighthouse n-Gons)

The Regular LH-n-Gons or Lighthouse n-Gons are n-Gons derived from the Lighthouse theorem.
If at each lighthouse the beams are equally spaced through 360°, the pairwise intersections of the beams form regular polygons: regular triangles for three beams, squares for four beams, etc.
The Lighthouse Theorem was discovered by R.K. Guy of the University of Calgary like he described at the Geometrical Internet Forum Hyacinthos Ref-11 message #2454. He wrote a paper on the Lighthouse Theorem relating it to theorems of Morley and Malfatti.
The Lighthouse Theorem describes how regular polygons can be constructed from the intersection of regular beams (rays) being emanated from two points (Lighthouses).

2P s iRg1 Regular LH n gons   01


The necessary elements for a regular Lighthouse Configuration are:
• Two lighthouses represented by two points A and B implying a baseline AB.
• Per lighthouse some initial ray represented by a line through the lighthouse point. The angle of these initial rays with baseline AB: a and b (called phases by Richard Guy at Ref-57) can be randomly chosen. Denote the init-rays as resp. a1, b1.
• A natural number n representing the number of rays per Lighthouse at equal angular distances, implying angles of p/n between the successive rays. These rays can be constructed by rotating the init-rays through p/n about A (or B), then rotating the new rays again through p/n about A (or B), etc. The new consecutive rays about A will be called a2, a3, etc. The new consecutive rays about B will be called b2, b3, etc. Rotating about A and B will be done in equal directions.
The Lighthouse Theorem states that the intersection points of n rays from 2 points (intersecting in n2 points) can be arranged in n regular n-gons.
The cyclic order of involved vertices per n-gon is:
          n-gon 1: (1,1), (2,2), … , (n,n)
          n-gon 2: (1,2), (2,3), … , (n,1)
          . . .
          . . .
          n-gon n: (1,n), (2,1), … , (n,n-1)
where (i,j) denotes the intersection point of ray ai and ray bj (i,j=1,2, … , n).


• Richard Guy proved in Ref-57 that the Morley Axes in a triangle (3-Line) can be constructed using the (general) Lighthouse Theorem.
• At nL-n-iL1 (nL-Morley's Axes) is described how in an n-Line, according to the extended lighthouse theorem, (semi-)regular polygons are formed, each with n axes meeting at angles i.p/n (i=1,2, … , n-1).
• Regular n-Gons also can be constructed from an n-Gon with random vertices using the Petr-Douglas-Neumann Theorem. See nG-n-iRg1.