Require Export D6_SumAnglesProp.
Require Export Omega.
Section DISTINCT_PARALLEL.

(* Specifications: *)
Lemma ThreeEmbeddedTriangles : forall A B C D E F G : Point,
Clockwise A B C ->
HalfLine A B D ->
HalfLine A C E ->
Angle A B C = Angle A D E ->
Angle A C B = Angle A E D ->
Between A D F ->
Between A E G ->
Distance D F = Distance A B ->
Distance E G = Distance A C ->
Angle A B C = Angle A F G /\ Angle A C B = Angle A G F.
Lemma EmbeddedSimilarTriangles : forall A B C : Point, forall n : nat,
Clockwise A B C ->
n > 0 ->
forall D E : Point,
HalfLine A B D ->
HalfLine A C E ->
Distance A D = DistTimesn n A B ->
Distance A E = DistTimesn n A C ->
Angle A B C = Angle A D E /\ Angle A C B = Angle A E D.
Lemma ExistsClockwiseParallelogramm : forall A B C D E : Point,
Parallelogramm A B C D ->
Between D E A ->
exists F : Point, Collinear C E F /\ Clockwise B A F.
Lemma ExistsClockwiseParallelogrammSym : forall A B C D E : Point,
Parallelogramm A B C D ->
Between A E C ->
exists F : Point, Collinear D E F /\ Clockwise B A F.
Lemma SecantParallelogrammSecant : forall A B : Point, forall F F' : Figure, forall D : Line F, forall D' : Line F',
Parallelogramm (LineA F D) (LineB F D) A B ->
F' A ->
~F' B ->
SecantLines F F' D D'.
Lemma ParallelogrammSuperimposed : forall A B C E : Point, forall F : Figure, forall D : Line F,
Parallelogramm A B C E ->
EquiDirected A B (LineA F D) (LineB F D) ->
F C ->
Superimposed F (Collinear C E).