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

theorem Th16:
  A,B,C is_a_triangle & angle(A,C,B) < PI &
  angle (E,B,A) = angle (C,B,A) / 3 &
  angle (B,A,E) = angle (B,A,C) / 3 &
  angle (A,C,F) = angle (A,C,B) / 3 &
  angle (F,A,C) = angle (B,A,C) / 3 &
  angle (C,B,G) = angle (C,B,A) / 3 &
  angle (G,C,B) = angle (A,C,B) / 3
  implies
  |.F-E.| = 4 * the_diameter_of_the_circumcircle(A,B,C)
     * sin (angle(A,C,B)/3) * sin (angle(C,B,A) /3) * sin (angle(B,A,C)/3) &
  |.G-F.| = 4 * the_diameter_of_the_circumcircle(C,A,B)
     * sin (angle(C,B,A)/3) * sin (angle(B,A,C) /3) * sin (angle(A,C,B)/3) &
  |.E-G.| = 4 * the_diameter_of_the_circumcircle(B,C,A)
     * sin (angle(B,A,C)/3) * sin (angle(A,C,B) /3) * sin (angle(C,B,A)/3)
  proof
    assume
A1: A,B,C is_a_triangle & angle(A,C,B) < PI &
    angle (E,B,A) = angle (C,B,A) / 3 &
    angle (B,A,E) = angle (B,A,C) / 3 &
    angle (A,C,F) = angle (A,C,B) / 3 &
    angle (F,A,C) = angle (B,A,C) / 3 &
    angle (C,B,G) = angle (C,B,A) / 3 &
    angle (G,C,B) = angle (A,C,B) / 3;
    then
A2: A,B,E is_a_triangle & A,F,C is_a_triangle &
    C,G,B is_a_triangle by Th13,Th14,Th15;
    now
      thus
A3:   C,A,B is_a_triangle & B,C,A is_a_triangle by A1,MENELAUS:15;
      thus C,A,F is_a_triangle & B,C,G is_a_triangle & B,E,A is_a_triangle
      by A2,MENELAUS:15;
      angle(A,C,B) <> 0 by A1,EUCLID10:30;
      then A,C,B are_mutually_distinct & 0 < angle(A,C,B) < PI
      by A1,A3,EUCLID_6:20,Th2;
      hence angle(C,B,A) < PI & angle(B,A,C) < PI by Th4;
    end;
    hence thesis by A1,A2,Th12;
  end;
