
theorem
  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)*
  4 * sin (angle(A,C,B)/3) * sin (PI/3 + angle(A,C,B)/3) *
  sin (PI/3 - angle(A,C,B)/3)
  proof
    let A,B,C be Point of TOP-REAL 2;
    assume
A1: A,B,C is_a_triangle;
    |.A-B.|=the_diameter_of_the_circumcircle(A,B,C)* sin (3*(angle(A,C,B)/3))
    by A1,Lm10;
    then |.A-B.|=the_diameter_of_the_circumcircle(A,B,C)*
    (4 * sin (angle(A,C,B)/3) * sin (PI/3 + (angle(A,C,B)/3)) *
    sin (PI/3 - (angle(A,C,B)/3))) by Thm18;
    hence thesis;
  end;
