
theorem Thm29:
  for A,B,C being Point of TOP-REAL 2 st A,B,C is_a_triangle holds
  the_diameter_of_the_circumcircle(A,B,C) = |.C-A.|/ sin(angle (C,B,A))
  proof
    let A,B,C be Point of TOP-REAL 2;
    assume A,B,C is_a_triangle; then
A2: A,B,C are_mutually_distinct by EUCLID_6:20;
    the_diameter_of_the_circumcircle(A,B,C)=
     |.A-B.|*|.B-C.|*|.C-A.|/2/(|.A-B.|*|.C-B.|*sin angle (C,B,A)/2)
    by EUCLID_6:5
    .=(|.A-B.|*|.B-C.|*|.C-A.|/2)/(1/2*(|.A-B.|*|.C-B.|*sin(angle (C,B,A))))
    .=((|.A-B.|*|.B-C.|*|.C-A.|/2)/(1/2))/(|.A-B.|*(|.C-B.| *
    sin(angle (C,B,A)))) by XCMPLX_1:78
    .=((|.A-B.|*|.B-C.|*|.C-A.|/2)/(1/2)/|.A-B.|)/(|.C-B.|*
    sin(angle (C,B,A))) by XCMPLX_1:78
    .=(|.A-B.|*(|.B-C.|*|.C-A.|)/|.A-B.|)/|.C-B.|/sin(angle (C,B,A))
    by XCMPLX_1:78
    .=(|.B-C.|*|.C-A.|)/|.C-B.|/sin(angle (C,B,A))
    by XCMPLX_1:89,A2,EUCLID_6:42
    .=((|.B-C.|*|.C-A.|)/|.B-C.|)/sin(angle (C,B,A)) by EUCLID_6:43
    .=|.C-A.|/sin(angle (C,B,A)) by XCMPLX_1:89,A2,EUCLID_6:42;
    hence thesis;
  end;
