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 Th54:
  A,B,C is_a_triangle & PI < angle(C,B,A) < 2 * PI implies
  the_diameter_of_the_circumcircle(A,B,C) < 0
  proof
    assume that
A1: A,B,C is_a_triangle and
A2: PI < angle(C,B,A) < 2 * PI;
    A,B,C are_mutually_distinct by A1,EUCLID_6:20;
    then
A3: |.C-A.| >=0 & |.C-A.| <> 0 by EUCLID_6:42;
    PI + 2 * PI * 0 < angle(C,B,A) & angle(C,B,A) < 2 * PI + 2 * PI * 0 by A2;
    then |.C-A.| / sin(angle(C,B,A)) < 0 by XREAL_1:142,A3,SIN_COS6:12;
    hence thesis by A1,EUCLID10:44;
end;
