Definition is_circumcenter' O A B C := 
  a_ratio O A B C (Py B C B * Py B A C / ((2*2*2*2*2) * (S A B C * S A B C)))
                            (Py A C A * Py A B C / ((2*2*2*2*2) * (S A B C * S A B C)))
                           (Py A B A * Py A C B / ((2*2*2*2*2) * (S A B C * S A B C))).Definition is_circumcenter O A B C := 
  a_ratio O A B C (Py B C B * Py B A C / (2*(Py A B A * Py A C A - Py B A C * Py B A C)))
                            (Py A C A * Py A B C /(2*(Py A B A * Py A C A - Py B A C * Py B A C)))
                           (Py A B A * Py A C B / (2*(Py A B A * Py A C A - Py B A C * Py B A C))).

Lemma is_circumcenter_non_zero' : forall H A B C, is_circumcenter H A B C -> (2*2*2*2*2) * (S A B C * S A B C) <> 0.