
theorem
  for A,B,C be Point of TOP-REAL 2 st
  A,B,C is_a_triangle & |( B - A , C - A )| = 0
  holds |.C-B.| * sin angle (C,B,A)=|.A-C.| or
  |.C-B.| * (- sin angle (C,B,A))= |.A-C.|
  proof
    let A,B,C be Point of TOP-REAL 2 such that
A1: A,B,C is_a_triangle and
A2: |( B - A , C - A )| = 0;
A3: A,B,C are_mutually_distinct by A1,EUCLID_6:20; then
A4: |. C - B .| * sin (angle(C,B,A)) = |. C - A .| * sin (angle(B,A,C))
    by EUCLID_6:6;
    sin angle(B,A,C)=1 or sin angle(B,A,C) = -1
    by A2,A3,SIN_COS:77,EUCLID_3:45;
    hence thesis by A4,EUCLID_6:43;
  end;
