(* elimination of area for euclidean constructions *)

(* Specifications: *)
Theorem elim_area_on_perp_d :
forall A B U V Y : Point, forall r: F,
on_perp_d Y U V r ->
S A B Y = S A B U - r / (2+2)  * Py4 U A V B.
Theorem elim_py_on_perp_d_left_right :
forall B U V Y : Point, forall r: F,
on_perp_d Y U V r ->
Py Y B Y = Py U B U - (2 + 2) * r * S U V B -
(2 + 2) * r * (0 - r / (2 + 2) * Py U V U + S U V B).
Theorem elim_area_on_foot :
forall A B P U V Y : Point,
on_foot Y P U V ->
S A B Y = (Py P U V * S A B V + Py P V U * S A B U) / (Py U V U).
Theorem elim_area_on_foot_invariant :
forall A B P U V Y : Point,
on_foot Y P U V ->
Py U V U <> 0.