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

theorem Th17:
  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.| = |.G-F.| & |.F-E.| = |.E-G.| & |.G-F.| = |.E-G.|
  proof
    assume 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
A1: |.F-E.| = the_diameter_of_the_circumcircle(A,B,C) * 4
    * sin (angle(A,C,B)/3) * sin (angle(C,B,A) /3) * sin (angle(B,A,C)/3) &
    |.G-F.| = the_diameter_of_the_circumcircle(C,A,B) * 4
    * sin (angle(C,B,A)/3) * sin (angle(B,A,C) /3) * sin (angle(A,C,B)/3) &
    |.E-G.| = the_diameter_of_the_circumcircle(B,C,A) * 4
    * sin (angle(B,A,C)/3) * sin (angle(A,C,B) /3) * sin (angle(C,B,A)/3)
    by Th16;
    the_diameter_of_the_circumcircle(A,B,C)
        = the_diameter_of_the_circumcircle(C,A,B) &
    the_diameter_of_the_circumcircle(A,B,C)
        = the_diameter_of_the_circumcircle(B,C,A) by EUCLID10:46;
    hence thesis by A1;
  end;
