Let A'B'C' be the Morley 1st Triangle.
Denote
A'b, A'c = the orthogonal projections of A' on CB', BC', resp.
B'c, B'a = the orthogonal projections of B' on AC', CA', resp.
C'a, C'b = the orthogonal projections of C' on BA', AB', resp.
1. A'b,A'c, B'c, B'a, C'a, C'b are con-conic (??).
2. The Euler Lines of A'AbAc, B'B'cB'a, C'C'aC'b are concurrent. (??)
Antreas P. Hatzipolakis, 11 Jan. 2013
Let A'B'C' be the Morley 1st Triangle.
Denote
A'b, A'c = the orthogonal projections of A' on AB', AC', resp.
B'c, B'a = the orthogonal projections of B' on BC', BA', resp.
C'a, C'b = the orthogonal projections of C' on CA', CB', resp.
1. A'b,A'c, B'c, B'a, C'a, C'b are concyclic (??).
2. The Euler Lines of A'AbAc, B'B'cB'a, C'C'aC'b are concurrent (??).
Antreas P. Hatzipolakis, 11 jan. 2012
Conjecture: The Euler lines of the triangles A'B"C", B'C"A", C'A"B" are concurrent, and also the Euler lines of the triangles A"B'C', B"C'A', C"A'B', if no one of the 6 triangles is degenerated.
A. P. Hatzipolakis, Hyacithos #21357
See the discussion in Hyacinthos #21341
Conjecture:
Let A'B'C' be the internal Morley triangle of triangle ABC and A"B"C" the (homothetic) Roussel equilateral triangle(*). The Euler Lines of A'B"C", B'C"A", C'A"B" are concurrent.
Biography of Nicolaides by Theodoros Passas published in the Supplement of the Bulletin of the Greek Mathematical Society #4 (May - June 1931)
Biography of Nicolaides by Michael Kasiouras published in Euclid, Supplement of the Bulletin of the Greek Mathematical Society, New Series, Vol. 4, #3(Nov. 1970)
See short biographies of Nicolaides
Greek Wikipedia HERE
By Spyros Zervos HERE
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In feminine geometry
all angles are tabooed;
But angling is woman's game,
when she is being wooed.
