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

theorem Th8:
  A,B,C is_a_triangle & A,F,C is_a_triangle & angle (C,F,A) < PI &
  angle (A,C,F) = angle (A,C,B)/3 & angle (F,A,C) = angle (B,A,C)/3 &
  angle (A,C,B)/3 + angle(B,A,C)/3 + angle(C,B,A)/3 = PI/3 &
  sin (PI/3 - angle(C,B,A)/3)<>0 implies
  |.A-F.| = 4 * the_diameter_of_the_circumcircle(A,B,C)
              * sin (angle(C,B,A) / 3) * sin (PI/3 + angle(C,B,A)/3)
              * sin (angle(A,C,B) / 3)
  proof
    assume that
A1: A,B,C is_a_triangle and
A2: A,F,C is_a_triangle and
A3: angle (C,F,A) < PI and
A4: angle (A,C,F) = angle (A,C,B)/3 and
A5: angle (F,A,C) = angle (B,A,C)/3 and
A6: angle (A,C,B)/3 + angle(B,A,C)/3 + angle(C,B,A)/3 = PI/3 and
A7: sin (PI/3 - angle(C,B,A)/3)<>0;
A8: |.A-F.| * sin (PI/3 - angle(C,B,A)/3) = |.A-C.| * sin (angle (A,C,B)/3)
                                            by A3,A4,A5,A2,A6,Th7
               .= |.C-A.| * sin (angle(A,C,B)/3) by EUCLID_6:43
               .= the_diameter_of_the_circumcircle(A,B,C) * sin angle(C,B,A)
                   * sin (angle(A,C,B)/3) by A1,EUCLID10:50;
     sin angle(C,B,A) = sin (3* (angle(C,B,A)/3))
                     .= 4 * sin (angle(C,B,A)/3) * sin (PI/3 + angle(C,B,A)/3)
                          * sin (PI/3 - angle(C,B,A)/3) by EUCLID10:29;
    then |.A-F.| =
    ((the_diameter_of_the_circumcircle(A,B,C) * 4 * sin (angle(C,B,A)/3) *
    sin (PI/3 + angle(C,B,A)/3) * sin (angle(A,C,B) /3)) *
    sin (PI/3 - angle(C,B,A)/3)) / sin (PI/3 - angle(C,B,A)/3)
    by A7,A8,XCMPLX_1:89;
    then
A9: |.A-F.| =
    (the_diameter_of_the_circumcircle(A,B,C) * 4 * sin (angle(C,B,A)/3) *
    sin (PI/3 + angle(C,B,A)/3) * sin (angle(A,C,B) /3)) *
    (sin (PI/3 - angle(C,B,A)/3) / sin (PI/3 - angle(C,B,A)/3));
    sin (PI/3 - angle(C,B,A)/3) / sin (PI/3 - angle(C,B,A)/3) = 1
    by A7,XCMPLX_1:60;
    hence thesis by A9;
  end;
