Definition square_dist A B :=
  let (Ac, _) := coordinates_of_point_F A in
  let (Ax, Ay) := Ac in
  let (Bc, _) := coordinates_of_point_F B in
  let (Bx, By) := Bc in
  (Ax - Bx) * (Ax - Bx) + (Ay - By) * (Ay - By).Definition twice_signed_area4 A B C D :=
  twice_signed_area A B C + twice_signed_area A C D.Definition Py A B C := square_dist A B + square_dist B C - square_dist A C.Definition Py4 A B C D := Py A B D - Py C B D.Definition AM_Cong A B C D := Py A B A =F= Py C D C.Definition AM_Col A B C := twice_signed_area A B C =F= 0.Definition AM_Per A B C := Py A B C =F= 0.Definition AM_Perp A B C D := Py4 A C B D =F= 0.Definition AM_Par A B C D := twice_signed_area4 A C B D =F= 0.Definition AM_on_foot Y P U V := AM_Perp Y P U V /\ AM_Col Y U V /\ U <> V.

Lemma chasles_ratios : forall A B C P Q, P <> Q -> ratio A B P Q + ratio B C P Q =F= ratio A C P Q.