theorem Th67:
  B <> C & the_foot_of_the_altitude(A,B,C),B,A is_a_triangle implies
  |.A-B.| * sin angle(A,B,the_foot_of_the_altitude(A,B,C))
      = |.the_foot_of_the_altitude(A,B,C)-A.| or
  |.A-B.| * (- sin angle(A,B,the_foot_of_the_altitude(A,B,C)))
      = |.the_foot_of_the_altitude(A,B,C)-A.|
  proof
    assume that
A1: B <> C and
A2: the_foot_of_the_altitude(A,B,C),B,A is_a_triangle;
    |(B-the_foot_of_the_altitude(A,B,C),
      A-the_foot_of_the_altitude(A,B,C))| = 0 by A1,Th39;
    hence thesis by A2,EUCLID10:32;
  end;
