theorem Th27:
  C in LSeg(A,B) & |.A-C.| = |.B-C.| implies half_length(A,B) = |.A-C.|
  proof
    assume that
A1: C in LSeg(A,B) and
A2: |.A-C.| = |.B-C.|;
A3: |.C-B.| = |.A-C.| by A2,EUCLID_6:43;
    half_length(A,B) = 1/2* (|.A-C.|+|.C-B.|) by A1,Th8
                    .= 1/2* (2*|.A-C.|) by A3;
    hence thesis;
  end;
