reserve A,B,C,D,E,F,G for Point of TOP-REAL 2;

theorem
  A,B,C is_a_triangle & angle(A,B,C) < PI &
  angle(E,C,A) = angle(B,C,A) / 3 &
  angle(C,A,E) = angle(C,A,B) / 3 &
  angle(A,B,F) = angle(A,B,C) / 3 &
  angle(F,A,B) = angle(C,A,B) / 3 &
  angle(B,C,G) = angle(B,C,A) / 3 &
  angle(G,B,C) = angle(A,B,C) / 3
  implies
  |.F-E.| = |.G-F.| & |.F-E.| = |.E-G.| & |.G-F.| = |.E-G.|
  proof
    assume
A1: A,B,C is_a_triangle & angle(A,B,C) < PI &
    angle(E,C,A) = angle(B,C,A) / 3 &
    angle(C,A,E) = angle(C,A,B) / 3 &
    angle(A,B,F) = angle(A,B,C) / 3 &
    angle(F,A,B) = angle(C,A,B) / 3 &
    angle(B,C,G) = angle(B,C,A) / 3 &
    angle(G,B,C) = angle(A,B,C) / 3;
    A,C,B is_a_triangle by A1,MENELAUS:15;
    hence thesis by A1,Th17;
  end;
