theorem Th56:
  A,B,C is_a_triangle & PI < angle(C,B,A) < 2 * PI &
  A in circle(a,b,r) & B in circle(a,b,r) & C in circle(a,b,r) implies
  the_diameter_of_the_circumcircle(A,B,C) = - 2 * r
  proof
    assume that
A1: A,B,C is_a_triangle and
A2: PI < angle(C,B,A) < 2 * PI and
A3: A in circle(a,b,r) & B in circle(a,b,r) & C in circle(a,b,r);
A4: the_diameter_of_the_circumcircle(A,B,C) = 2 * r or
    the_diameter_of_the_circumcircle(A,B,C) = - 2 * r by A1,A3,Th52;
    r > 0 by A1,A3,EUCLID10:37;
    hence thesis by A1,A2,Th54,A4;
  end;
