(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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Local Open Scope R_scope.
(** * Distance *)
Definition dist_euc (x0 y0 x1 y1:R) : R :=
sqrt (Rsqr (x0 - x1) + Rsqr (y0 - y1)).

Lemma law_cosines :
forall x0 y0 x1 y1 x2 y2 ac:R,
let a := dist_euc x1 y1 x0 y0 in
let b := dist_euc x2 y2 x0 y0 in
let c := dist_euc x2 y2 x1 y1 in
a * c * cos ac = (x0 - x1) * (x2 - x1) + (y0 - y1) * (y2 - y1) ->
Rsqr b = Rsqr c + Rsqr a - 2 * (a * c * cos ac).