Require Export C2_Entre_Prop.
Section CONGRUENT_TRIANGLES.
Definition CongruentTriangles (A B C A' B' C' : Point) :=
Distance A B = Distance A' B' /\
Distance B C = Distance  B' C' /\
Distance C A = Distance C' A'.

(* Specifications: *)
Lemma CongruentTrianglesSym : forall A B C A' B' C' : Point,
CongruentTriangles A B C A' B' C' ->
CongruentTriangles A' B' C' A B C.
Lemma CongruentTrianglesBAC : forall A B C A' B' C' : Point,
CongruentTriangles A B C A' B' C' ->
CongruentTriangles B A C B' A' C'.
Lemma CongruentTrianglesACB : forall A B C A' B' C' : Point,
CongruentTriangles A B C A' B' C' ->
CongruentTriangles A C B A' C' B'.
Lemma CongruentTrianglesCBA : forall A B C A' B' C' : Point,
CongruentTriangles A B C A' B' C' ->
CongruentTriangles C B A C' B' A'.
Lemma CongruentTrianglesBCA : forall A B C A' B' C' : Point,
CongruentTriangles A B C A' B' C' ->
CongruentTriangles B C A B' C' A'.
Lemma CongruentTrianglesCAB : forall A B C A' B' C' : Point,
CongruentTriangles A B C A' B' C' ->
CongruentTriangles C A B C' A' B'.
Lemma CongruentTrianglesAB : forall A B C A' B' C' : Point,
CongruentTriangles A B C A' B' C' ->
Distance A B = Distance A' B'.
Lemma CongruentTrianglesBC : forall A B C A' B' C' : Point,
CongruentTriangles A B C A' B' C' ->
Distance B C = Distance B' C'.
Lemma CongruentTrianglesCA : forall A B C A' B' C' : Point,
CongruentTriangles A B C A' B' C' ->
Distance C A = Distance C' A'.
Lemma CongruentTrianglesA : forall A B C A' B' C' : Point,
CongruentTriangles A B C A' B' C' ->
A <> B ->
A <> C ->
Angle B A C = Angle B' A' C'.
Lemma CongruentTrianglesB : forall A B C A' B' C' : Point,
CongruentTriangles A B C A' B' C' ->
B <> A ->
B <> C ->
Angle A B C = Angle A' B' C'.
Lemma CongruentTrianglesC : forall A B C A' B' C' : Point,
CongruentTriangles A B C A' B' C' ->
C <> A ->
C <> B ->
Angle A C B = Angle A' C' B'.
Lemma CongruentTrianglesSASA : forall A B C A' B' C' : Point,
Distance A B = Distance A' B' ->
Distance A C = Distance  A' C' ->
Angle B A C = Angle B' A'  C' ->
A <> B ->
A <> C ->
CongruentTriangles A B C A' B' C'.
Lemma CongruentTrianglesSASB : forall A B C A' B' C' : Point,
Distance B A = Distance B' A' ->
Distance B C = Distance  B' C' ->
Angle A B C = Angle A' B' C' ->
B <> A ->
B <> C ->
CongruentTriangles A B C A' B' C'.
Lemma CongruentTrianglesSASC : forall A B C A' B' C' : Point,
Distance C A = Distance C' A'  ->
Distance C B = Distance  C' B'  ->
Angle A C B = Angle A' C' B' ->
C <> A ->
C <> B ->
CongruentTriangles A B C A' B' C'.