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 Th71:
  B,C,A is_a_triangle & angle(B,C,A) < PI & D,C,A is_a_triangle &
  angle(C,D,A)=PI/2 implies
  |.D-C.| = |.A-B.| * sin angle(A,B,C) / sin (angle(A,B,C) + angle(C,A,B))
             * sin angle(D,A,C)
  proof
    assume that
A1: B,C,A is_a_triangle and
A2: angle(B,C,A) < PI and
A3: D,C,A is_a_triangle and
A4: angle(C,D,A)=PI/2;
    |.D-C.| = |.A-C.| * sin angle(D,A,C) by A3,A4,EUCLID10:34
           .= |.B-A.| * sin angle(A,B,C) / sin (angle(A,B,C) + angle(C,A,B))
                      * sin angle(D,A,C) by A1,A2,Th63;
    hence thesis by EUCLID_6:43;
  end;
