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;
