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

theorem Th12:
  A,B,C is_a_triangle & A,B,E is_a_triangle &
  angle (E,B,A) = angle (C,B,A) / 3 & angle (B,A,E) = angle (B,A,C) / 3 &
  A,F,C is_a_triangle &
  angle (A,C,F) = angle (A,C,B)/3 & angle (F,A,C) = angle (B,A,C)/3 &
  angle(A,C,B) < PI
  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)
  proof
    assume that
A1: A,B,C is_a_triangle and
A2: A,B,E is_a_triangle and
A3: angle (E,B,A) = angle (C,B,A) / 3 and
A4: angle (B,A,E) = angle (B,A,C) / 3 and
A5: A,F,C is_a_triangle and
A6: angle (A,C,F) = angle (A,C,B)/3 and
A7: angle (F,A,C) = angle (B,A,C)/3 and
A8: angle(A,C,B) < PI;
A9: A,C,B is_a_triangle by A1,MENELAUS:15;
    then
A10: sin (PI/3 - angle(A,C,B)/3)<>0 by Th11;
    C,B,A is_a_triangle by A1,MENELAUS:15;
    then
A11: sin (PI/3 - angle(C,B,A)/3)<>0 by Th11;
A12: A,C,B are_mutually_distinct by A9,EUCLID_6:20;
    then angle (A,C,B) + angle(C,B,A) + angle(B,A,C) = PI
    by A8,EUCLID_3:47;
    then
A13: angle (A,C,B)/3 + angle(B,A,C)/3 + angle(C,B,A)/3 = PI/3;
    angle(A,C,B) <> 0 by A1,EUCLID10:30;
    then
A14: 0 < angle(A,C,B) by Th2;
    then
A15: angle (C,B,A) < PI by A12,A8,Th4;
    angle(A,C,B) <> 0 by A1,EUCLID10:30;
    then
A16: 0 < angle(A,C,B) < PI & A,C,B are_mutually_distinct
    by A9,EUCLID_6:20,A8,Th2;
    then angle(B,A,C) < PI by Th4;
    then
A17: angle(B,A,C)/3 < PI/3 & PI/3 < PI
    by XREAL_1:74,XREAL_1:221,COMPTRIG:5;
A18: E,A,B is_a_triangle &
    B,A,E is_a_triangle & E,B,A is_a_triangle by A2,MENELAUS:15;
    then
A19: B,A,E are_mutually_distinct & E,B,A are_mutually_distinct &
    angle(E,B,A) <> 0 & angle(B,A,E) <> 0 by EUCLID_6:20,EUCLID10:30;
    0 < angle(B,A,E) < PI by A17,A4,XXREAL_0:2,A19,Th2;
    then
A20: angle(A,E,B) < PI by A19,Th4;
    F,A,C is_a_triangle by A5,MENELAUS:15;
    then
A21: F,A,C are_mutually_distinct & angle(F,A,C) <> 0
    by EUCLID_6:20,EUCLID10:30;
    then 0 < angle(F,A,C) < PI by A17,A7,XXREAL_0:2,Th2;
    then
A22: angle(C,F,A) < PI by A21,Th4;
A23: F,A,E is_a_triangle
    proof
      now
        E<>F
        proof
          assume
A24:      E=F;
          per cases by EUCLID_6:4;
          suppose
A25:        angle(B,A,E) + angle(E,A,C) = angle(B,A,C);
A26:        0 < angle(B,A,C) by A16,Th4;
            then
A27:        angle(B,A,C) / angle(B,A,C) = 1 by XCMPLX_1:60;
            angle(B,A,C) / angle(B,A,C) = 2/3 * angle(B,A,C) / angle(B,A,C)
                                              by A24,A7,A4,A25
                                       .= 2/3 by A27;
            hence contradiction by A26,XCMPLX_1:60;
          end;
          suppose
A28:        angle(B,A,E)+angle(E,A,C) = angle(B,A,C) + 2*PI;
A29:        0 < angle(B,A,C) by A16,Th4;
            2*PI = -1/3 * angle(B,A,C) by A24,A7,A4,A28;
            hence contradiction by A29,COMPTRIG:5;
          end;
        end;
        hence F,A,E are_mutually_distinct by A21,A19;
        now
          hereby
            assume
A30:        angle(F,A,E)=PI;
A31:        angle(C,A,F)+angle(F,A,E)=angle(C,A,E) or
            angle(C,A,F)+angle(F,A,E)=angle(C,A,E)+2*PI by EUCLID_6:4;
            per cases by EUCLID_6:4;
            suppose angle(C,A,E)+angle(E,A,B)=angle(C,A,B);
              then per cases by A31,A30;
              suppose
A32:            angle(C,A,F) + PI + angle(E,A,B) = angle(C,A,B);
A33:            0 < angle(B,A,C) by A16,Th4;
A34:            angle(C,A,F) + PI + angle(E,A,B) = 2*PI - angle(B,A,C)
                by A32,A1,EUCLID10:31;
A35:            angle(C,A,F) = 2*PI - angle(B,A,C)/3 by A7,A5,EUCLID10:31;
                angle(E,A,B) = 2*PI - angle(B,A,E) = 2*PI - angle(B,A,C)/3
                               by A4,A2,EUCLID10:31;
                then 3*PI = -1/3 * angle(B,A,C) by A34,A35;
                hence contradiction by A33,COMPTRIG:5;
              end;
              suppose
A36:            angle(C,A,F) - PI +angle(E,A,B)=angle(C,A,B);
A37:            0 < angle(B,A,C) by A16,Th4;
A38:            angle(C,A,F) - PI + angle(E,A,B) = 2*PI - angle(B,A,C)
                           by A36,A1,EUCLID10:31;
A39:            angle(C,A,F) = 2*PI - angle(F,A,C) = 2*PI - angle(B,A,C)/3
                           by A7,A5,EUCLID10:31;
                angle(E,A,B) = 2*PI - angle(B,A,C)/3 by A4,A2,EUCLID10:31;
                hence contradiction by A37,COMPTRIG:5,A38,A39;
              end;
            end;
            suppose angle(C,A,E)+angle(E,A,B)=angle(C,A,B)+2*PI;
              then per cases by A31,A30;
              suppose
A40:            angle(C,A,F)+ PI +angle(E,A,B)=angle(C,A,B)+2*PI;
A41:            0 < angle(B,A,C) by A16,Th4;
A42:            angle(C,A,F) + PI + angle(E,A,B)
                   = 2*PI - angle(B,A,C) + 2*PI by A40,A1,EUCLID10:31;
A43:            angle(C,A,F) = 2*PI - angle(B,A,C)/3 by A7,A5,EUCLID10:31;
                angle(E,A,B) = 2*PI - angle(B,A,C)/3 by A4,A2,EUCLID10:31;
                hence contradiction by A42,A43,A41,COMPTRIG:5;
              end;
              suppose
A44:            angle(C,A,F)- PI +angle(E,A,B)=angle(C,A,B)+2*PI;
A45:            angle(C,A,F) - PI + angle(E,A,B)
                  = 2*PI - angle(B,A,C) + 2*PI by A44,A1,EUCLID10:31;
A46:            angle(C,A,F) = 2*PI - angle(B,A,C)/3 by A7,A5,EUCLID10:31;
                angle(E,A,B) = 2*PI - angle(B,A,C)/3 by A4,A2,EUCLID10:31;
                then 3 * PI = 3 * 1/3 * angle(B,A,C) & 2 * PI + 0 < 2 * PI + PI
                by A45,A46,XREAL_1:6,COMPTRIG:5;
                hence contradiction by Th2;
              end;
            end;
          end;
          hereby
            assume angle(A,E,F)=PI;
            then
A47:        E in LSeg(A,F) & E<> A by A19,EUCLID_6:11;
A48:        angle(C,A,F) + angle(F,A,B)=angle(C,A,B) or
            angle(C,A,F) + angle(F,A,B)=angle(C,A,B) + 2*PI by EUCLID_6:4;
A49:        angle(C,A,F) = 2*PI - angle(B,A,C)/3 by A7,A5,EUCLID10:31;
            angle(F,A,B) = angle(E,A,B) by A47,EUCLID_6:9
                        .= 2*PI - angle(B,A,C)/3 by A4,A2,EUCLID10:31;
            then per cases by A1,EUCLID10:31,A48,A49;
            suppose
A50:          2*PI - angle(B,A,C)/3 + 2*PI -angle(B,A,C)/3
                 = 2*PI - angle(B,A,C);
              0 < angle(B,A,C) by A16,Th4;
              hence contradiction by A50,COMPTRIG:5;
            end;
            suppose 2*PI - angle(B,A,C)/3 + 2*PI -angle(B,A,C)/3 =
              2*PI - angle(B,A,C) + 2*PI;
              hence contradiction by A16,Th4;
            end;
          end;
          hereby
            assume angle(E,F,A)=PI;
            then
A51:        F in LSeg(E,A) & F <> A by A21,EUCLID_6:11;
A52:        angle(C,A,F) + angle(F,A,B)=angle(C,A,B) or
            angle(C,A,F) + angle(F,A,B)=angle(C,A,B) + 2*PI by EUCLID_6:4;
A53:        angle(C,A,F) = 2*PI - angle(B,A,C)/3 by A5,A7,EUCLID10:31;
            angle(F,A,B) = angle(E,A,B) by A51,EUCLID_6:9
            .= 2*PI - angle(B,A,C)/3 by A4,A2,EUCLID10:31;
            then per cases by A1,EUCLID10:31,A52,A53;
            suppose
A54:          2*PI - angle(B,A,C)/3 + 2*PI -angle(B,A,C)/3
                 = 2*PI - angle(B,A,C);
              0 < angle(B,A,C) by A16,Th4;
              hence contradiction by A54,COMPTRIG:5;
            end;
            suppose
              2*PI - angle(B,A,C)/3 + 2*PI -angle(B,A,C)/3
                = 2*PI - angle(B,A,C) + 2*PI;
              hence contradiction by A16,Th4;
            end;
          end;
        end;
        hence angle(F,A,E)<>PI & angle(A,E,F)<>PI & angle(E,F,A)<>PI;
      end;
      hence thesis by EUCLID_6:20;
    end;
A55: - the_diameter_of_the_circumcircle(C,B,A)
    = the_diameter_of_the_circumcircle(A,B,C) by A1,EUCLID10:49;
    set lambda = (- the_diameter_of_the_circumcircle(C,B,A) * 4
    * sin (angle(A,C,B)/3) * sin (angle(C,B,A) /3));
A56: |.A-E.| =
    (- the_diameter_of_the_circumcircle(C,B,A) * 4 * sin (angle(A,C,B)/3) *
    sin (PI/3 + angle(A,C,B)/3) * sin (angle(C,B,A) /3))
    by A1,A2,A3,A4,A15,A20,A10,EUCLID10:56
    .= lambda * sin(PI/3 + angle(A,C,B)/3);
A57: |.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))
    by A1,A5,A6,A7,A11,A13,A22,Th8
    .= lambda * sin(PI/3 + angle(C,B,A)/3) by A55;
A58: C,A,B is_a_triangle by A1,MENELAUS:15;
    then
A59: (PI/3 + angle(A,C,B)/3) + (PI/3 + angle(C,B,A)/3) + angle(E,A,F) = PI
    by Th10,A5,A23,A18,A4,A7,A8;
A60: |.F-E.|^2 =
    |.A-E.|^2 + |.A-F.|^2 - 2*|.A-E.|*|.A-F.|*cos angle (E,A,F) by Th3
    .= lambda^2 * ((sin(PI/3 + angle(A,C,B)/3))^2
           + (sin(PI/3 + angle(C,B,A)/3))^2
           - 2 * sin(PI/3 + angle(A,C,B)/3)
           * sin(PI/3 + angle(C,B,A)/3) * cos angle(E,A,F))
    by A56,A57
    .= lambda^2 * (sin angle(E,A,F))^2 by A59,EUCLID10:25
    .= lambda^2 * (sin (angle(B,A,C)/3))^2 by Th9,A58,A5,A23,A18,A4,A7
    .= (lambda * sin(angle(B,A,C)/3))^2;
    now
A61:  lambda = (- the_diameter_of_the_circumcircle(C,B,A)) * 4
                * sin (angle(A,C,B)/3) * sin (angle(C,B,A) /3)
            .= the_diameter_of_the_circumcircle(A,B,C) * 4
                * sin (angle(A,C,B)/3) * sin (angle(C,B,A) /3)
                 by A1,EUCLID10:49;
      now
        0 <= angle(C,B,A) < PI by A14,A12,A8,Th4;
        then
A62:    0 <= angle(C,B,A)/3 < PI/3 by XREAL_1:74;
        PI/3 < PI/1 by COMPTRIG:5,XREAL_1:76;
        then 2*PI*0 <= angle(C,B,A)/3 < PI+2*PI*0 by A62,XXREAL_0:2;
        hence sin (angle(C,B,A)/3) >=0 by SIN_COS6:16;
        0 <= angle(A,C,B) < PI by A8,Th2;
        then
A63:    0 <= angle(A,C,B)/3 < PI/3 by XREAL_1:74;
        PI/3 < PI/1 by COMPTRIG:5,XREAL_1:76;
        then 2*PI*0 <= angle(A,C,B)/3 < PI+2*PI*0 by A63,XXREAL_0:2;
        hence 0 <= sin (angle(A,C,B)/3) by SIN_COS6:16;
        thus 0 <= the_diameter_of_the_circumcircle(A,B,C) by A15,Th6;
      end;
      hence 0 <= lambda by A61;
      now
        A,C,B is_a_triangle by A1,MENELAUS:15;
        hence A,C,B are_mutually_distinct by EUCLID_6:20;
        angle(A,C,B) <> 0 by A1,EUCLID10:30;
        hence 0 < angle(A,C,B) < PI by A8,Th2;
      end;
      then 0 <= angle(B,A,C) < PI by Th4;
      then
A64:  0 <= angle(B,A,C)/3 < PI/3 by XREAL_1:74;
      PI/3 < PI/1 by COMPTRIG:5,XREAL_1:76;
      then  2*PI*0 <= angle(B,A,C)/3 < PI+2*PI*0 by A64,XXREAL_0:2;
      hence 0 <= sin (angle(B,A,C)/3) by SIN_COS6:16;
    end;
    then
A65: sqrt ((lambda * sin(angle(B,A,C)/3))^2) = lambda
    * sin(angle(B,A,C)/3) by SQUARE_1:22;
    lambda = (- the_diameter_of_the_circumcircle(C,B,A)) * 4
      * sin (angle(A,C,B)/3) * sin (angle(C,B,A) /3)
          .= the_diameter_of_the_circumcircle(A,B,C) * 4 * sin (angle(A,C,B)/3)
      * sin (angle(C,B,A) /3) by A1,EUCLID10:49;
    hence thesis by A60,SQUARE_1:22,A65;
  end;
