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;
