theorem Th57:
  A,B,C is_a_triangle & 0 < angle(C,B,A) < PI &
  A in circle(a,b,r) & B in circle(a,b,r) & C in circle(a,b,r)
  implies
  |.A-B.| = 2 * r * sin angle(A,C,B) &
  |.B-C.| = 2 * r * sin angle(B,A,C) &
  |.C-A.| = 2 * r * sin angle(C,B,A)
  proof
    assume that
A1: A,B,C is_a_triangle and
A2: 0 < angle(C,B,A) < PI and
A3: A in circle(a,b,r) & B in circle(a,b,r) & C in circle(a,b,r);
    the_diameter_of_the_circumcircle(A,B,C) = 2 * r by A1,A2,A3,Th55;
    hence thesis by A1,EUCLID10:50;
  end;
