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 Th66:
  A,B,C is_a_triangle & angle(C,A,B) = 3/2*PI implies
  the_length_of_the_altitude(C,A,B) = |.A-B.| * tan angle(C,B,A)
  proof
    assume that
A1: A,B,C is_a_triangle and
A2: angle(C,A,B) = 3/2*PI;
A3: A,B,C are_mutually_distinct by A1,EUCLID_6:20;
    then |(C-A,B-A)| = 0 by A2,EUCLID_3:45;
    then
A4: |(A-B,A-C)| = 0 by Th10;
    angle(B,A,C) = 2*PI - 3/2*PI by A1,A2,EUCLID10:31
                .= PI/2;
    then tan angle(C,B,A) * |.A-B.| = |.A-C.|/|.A-B.|*|.A-B.| by A1,EUCLID10:35
                              .= |.A-C.| by A3,EUCLID_6:42,XCMPLX_1:87
                              .= |.C-A.| by EUCLID_6:43;
    then |.A-B.| * tan angle(C,B,A)
                  = |.the_foot_of_the_altitude(C,A,B) - C.| by A4,A3,Th47
                 .= |.C - the_foot_of_the_altitude(C,A,B).| by EUCLID_6:43;
    hence thesis by A3,Def3;
  end;
