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 Th70:
  A,C,B is_a_triangle & angle(A,C,B) < PI & D,A,C is_a_triangle &
  angle(A,D,C)=PI/2 implies
  |.D-C.| = |.A-B.| * sin angle(C,B,A) / sin (angle(B,A,C) + angle(C,B,A))
                    * sin angle(C,A,D)
  proof
    assume that
A1: A,C,B is_a_triangle and
A2: angle(A,C,B) < PI and
A3: D,A,C is_a_triangle and
A4: angle(A,D,C)=PI/2;
    |.D-C.| = |.C-A.| * sin angle(C,A,D) by A3,A4,EUCLID10:34
           .= |.A-C.| * sin angle(C,A,D) by EUCLID_6:43
           .= |.A-B.| * sin angle(C,B,A) / sin (angle(B,A,C) + angle(C,B,A))
                      * sin angle(C,A,D) by A1,A2,Th64;
    hence thesis;
  end;
