
theorem Thm21:
  for A,B,C be Point of TOP-REAL 2 st
  A,B,C is_a_triangle & angle(B,A,C)=PI/2 holds
  |.C-B.| * sin angle (C,B,A) = |.A-C.| &
  |.C-B.| * sin angle (A,C,B) = |.A-B.| &
  |.C-B.| * cos angle (C,B,A) = |.A-B.| &
  |.C-B.| * cos angle (A,C,B) = |.A-C.|
  proof
    let A,B,C be Point of TOP-REAL 2 such that
A1: A,B,C is_a_triangle and
A2: angle(B,A,C) = PI/2;
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;
A5: |. B - A .| * sin angle(B,A,C)=|.B - C.| * sin angle(A,C,B)
    by A3,EUCLID_6:6; thus
A6: |.C-B.| * sin angle (C,B,A)=|.A-C.| by A2,A4,SIN_COS:77,EUCLID_6:43;
    |. A - B .| * sin angle(B,A,C)=|.B- C.| * sin angle(A,C,B)
    by A5,EUCLID_6:43;
    hence
A7: |.C-B.| * sin angle (A,C,B)=|.A-B.| by A2,SIN_COS:77,EUCLID_6:43;
    angle(B,A,C)+angle(A,C,B)+angle(C,B,A)=PI by A2,A3,COMPTRIG:5,EUCLID_3:47;
    then sin angle(A,C,B)=sin (PI/2-angle(C,B,A)) by A2;
    hence |.C-B.| * cos angle (C,B,A)=|.A-B.| by A7,SIN_COS:79;
    angle(B,A,C)+angle(A,C,B)+angle(C,B,A)=PI by A2,A3,COMPTRIG:5,EUCLID_3:47;
    then sin angle(C,B,A)=sin (PI/2-angle(A,C,B)) by A2;
    hence |.C-B.| * cos angle (A,C,B)=|.A-C.| by A6,SIN_COS:79;
  end;
