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 & 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
  |.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: 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);
    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;
    PI + 2 * PI * 0 < angle(C,B,A) & angle(C,B,A) < 2 * PI + 2 * PI * 0 by A2;
    then
A5: sin(angle(C,B,A)) < 0 by SIN_COS6:12;
    PI + 2 * PI * 0 < angle(B,A,C) &
    angle(B,A,C) < 2 * PI + 2 * PI * 0 by A2,A4,EUCLID11:8,EUCLID11:2;
    then
A6: sin(angle(B,A,C)) < 0 by SIN_COS6:12;
    PI + 2 * PI * 0 < angle(A,C,B) & angle(A,C,B) < 2 * PI + 2 * PI * 0
      by A2,A4,EUCLID11:2,8; then
A7: sin(angle(A,C,B)) < 0 by SIN_COS6:12;
    |.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) by A1,A2,A3,Th58;
    then |.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);
    hence thesis by A5,A6,A7,XCMPLX_1:89;
  end;
