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
  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.| / sin angle(A,C,B) = 2 * r &
  |.B-C.| / sin angle(B,A,C) = 2 * r &
  |.C-A.| / sin angle(C,B,A) = 2 * r
  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);
    A,B,C are_mutually_distinct by A1,EUCLID_6:20; then
A4: C,B,A are_mutually_distinct & B,A,C are_mutually_distinct;
    2 * PI * 0 < angle(C,B,A) & angle(C,B,A) < PI + 2 * PI * 0 by A2; then
A5: sin(angle(C,B,A)) > 0 by SIN_COS6:11;
    2 * PI * 0 < angle(B,A,C) & angle(B,A,C) < PI + 2 * PI * 0
      by A2,A4,EUCLID11:5; then
A6: sin(angle(B,A,C)) > 0 by SIN_COS6:11;
    2 * PI * 0 < angle(A,C,B) & angle(A,C,B) < PI + 2 * PI * 0
      by A2,A4,EUCLID11:5; then
A7: sin(angle(A,C,B)) > 0 by SIN_COS6:11;
    |.A-B.| / sin angle(A,C,B) = (2 * r) * sin angle(A,C,B) / sin angle(A,C,B)
    &
    |.B-C.| / sin angle(B,A,C) = (2 * r) * sin angle(B,A,C) / sin angle(B,A,C)
    &
    |.C-A.| / sin angle(C,B,A) = (2 * r) * sin angle(C,B,A) / sin angle(C,B,A)
       by A1,A2,A3,Th57;
    hence thesis by A5,A6,A7,XCMPLX_1:89;
  end;
