reserve n for Nat,
        lambda,lambda2,mu,mu2 for Real,
        x1,x2 for Element of REAL n,
        An,Bn,Cn for Point of TOP-REAL n,
        a for Real;
 reserve Pn,PAn,PBn for Element of REAL n,
         Ln for Element of line_of_REAL n;
reserve A,B,C for Point of TOP-REAL 2;
reserve x,y,z,y1,y2 for Element of REAL 2;
reserve L,L1,L2,L3,L4 for Element of line_of_REAL 2;
reserve D,E,F for Point of TOP-REAL 2;
reserve b,c,d,r,s for Real;

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;
