
theorem Thm34:
  for A,B,C be Point of TOP-REAL 2 st
  A,B,C is_a_triangle holds
  |.A-B.|= the_diameter_of_the_circumcircle(A,B,C)*sin angle (A,C,B) &
  |.B-C.|= the_diameter_of_the_circumcircle(A,B,C)*sin angle (B,A,C) &
  |.C-A.|= the_diameter_of_the_circumcircle(A,B,C)*sin angle (C,B,A)
  proof
    let A,B,C be Point of TOP-REAL 2;
    assume
A1: A,B,C is_a_triangle; then
A2: B,C,A is_a_triangle & C,A,B is_a_triangle by MENELAUS:15;
A3: |.B-C.|=the_diameter_of_the_circumcircle(B,C,A)*sin angle (B,A,C)
    by A2,Lm10;
    thus|.A-B.|= the_diameter_of_the_circumcircle(A,B,C)*sin angle (A,C,B)
    by A1,Lm10;
    thus |.B-C.|=the_diameter_of_the_circumcircle(A,B,C)*sin angle (B,A,C)
    by A3,Thm27;
    |.C-A.|=the_diameter_of_the_circumcircle(C,A,B)*sin angle (C,B,A)
    by A2,Lm10;
    hence |.C-A.|=the_diameter_of_the_circumcircle(A,B,C)*sin angle (C,B,A)
    by Thm27;
  end;
