Are you able to help with investigating the Lighthouse Theorem for values of n > 4 ? For information about the theorem and the important Lighthouse Duplication Theorem see the article in the Amer Math Monthly. Alternatively, I can send a copy. For n = 5, it seems that two applications of the Lighthouse Duplication theorem should lead to sets of homothetic regular pentagons which are related to quinquisectors of two angles (of a triangle? of a pentagon?) For n = 9, there can be three applications of the Lighthouse Duplication theorem. In any case a subset of the novisectors are the trisectors of Morley's theorem. Moreover the remaining novisectors Will lead to 18 Morley triangles for each of 3 x 18 triangles in the same construction. If XYZ is a Morley triangle of ABC, then the triangles I mean are BCX, CAY, ABZ, but there are 3 x 18 other triangles, AYZ, BZX, CXY, which may be worth examining. To what extent are the resulting equilateral triangles homothetic? Best wishes. R.