reserve n for Nat;
reserve i for Integer;
reserve r,s,t for Real;
reserve An,Bn,Cn,Dn for Point of TOP-REAL n;
reserve L1,L2 for Element of line_of_REAL n;
reserve A,B,C for Point of TOP-REAL 2;
reserve D for Point of TOP-REAL 2;
reserve a,b,c,d for Real;

theorem Th64:
  A,C,B is_a_triangle & angle(A,C,B) < PI implies
  |.A-C.| = |.A-B.| * sin angle(C,B,A) / sin (angle(B,A,C) + angle(C,B,A))
  proof
    assume that
A1: A,C,B is_a_triangle and
A2: angle(A,C,B) < PI;
A3: A,C,B are_mutually_distinct by A1,EUCLID_6:20;
A4: |.A-C.| = |.A-B.| * sin angle(C,B,A) / sin angle(A,C,B)
    proof
      |.A - B.| * sin angle(C,B,A) / sin angle(A,C,B)
             = |.A - C.| * sin angle(A,C,B) / sin angle (A,C,B)
                   by A3,EUCLID_6:6
            .= |.A - C.| * (sin angle(A,C,B) / sin angle (A,C,B))
            .= |.A - C.| * 1 by A1,Th22,XCMPLX_1:60
            .= |.A - C.|;
      hence thesis;
    end;
    angle(A,C,B) = PI - (angle(C,B,A) + angle(B,A,C))
    proof
      A,C,B are_mutually_distinct by A1,EUCLID_6:20;
      then angle(A,C,B) + angle (C,B,A) + angle (B,A,C) = PI by A2,EUCLID_3:47;
      hence thesis;
    end;
    hence thesis by A4,EUCLID10:1;
  end;
