# Library GeoCoq.Elements.OriginalProofs.proposition_37

Require Export GeoCoq.Elements.OriginalProofs.lemma_triangletoparallelogram.

Require Export GeoCoq.Elements.OriginalProofs.lemma_PGrotate.

Require Export GeoCoq.Elements.OriginalProofs.proposition_35.

Section Euclid.

Context `{Ax:area}.

Lemma proposition_37 :

∀ A B C D,

Par A D B C →

ET A B C D B C.

Proof.

intros.

assert (Par B C A D) by (conclude lemma_parallelsymmetric).

assert (Par C B A D) by (forward_using lemma_parallelflip).

assert (eq A A) by (conclude cn_equalityreflexive).

assert (eq D D) by (conclude cn_equalityreflexive).

assert (Col A D A) by (conclude_def Col ).

assert (Col A D D) by (conclude_def Col ).

let Tf:=fresh in

assert (Tf:∃ E, (PG A E B C ∧ Col A D E)) by (conclude lemma_triangletoparallelogram);destruct Tf as [E];spliter.

let Tf:=fresh in

assert (Tf:∃ F, (PG D F B C ∧ Col A D F)) by (conclude lemma_triangletoparallelogram);destruct Tf as [F];spliter.

assert (PG E B C A) by (conclude lemma_PGrotate).

assert (PG F B C D) by (conclude lemma_PGrotate).

assert (Col D A F) by (forward_using lemma_collinearorder).

assert (Col D A E) by (forward_using lemma_collinearorder).

assert (nCol C A D) by (forward_using lemma_parallelNC).

assert (neq A D) by (forward_using lemma_NCdistinct).

assert (neq D A) by (conclude lemma_inequalitysymmetric).

assert (Col A F E) by (conclude lemma_collinear4).

assert (Col E A D) by (forward_using lemma_collinearorder).

assert (Col E A F) by (forward_using lemma_collinearorder).

assert (EF E B C A F B C D) by (conclude proposition_35).

assert (Cong_3 B E A A C B) by (conclude proposition_34).

assert (Cong_3 B F D D C B) by (conclude proposition_34).

let Tf:=fresh in

assert (Tf:∃ M, (BetS E M C ∧ BetS B M A)) by (conclude lemma_diagonalsmeet);destruct Tf as [M];spliter.

let Tf:=fresh in

assert (Tf:∃ m, (BetS F m C ∧ BetS B m D)) by (conclude lemma_diagonalsmeet);destruct Tf as [m];spliter.

assert (Col B M A) by (conclude_def Col ).

assert (Col B m D) by (conclude_def Col ).

assert (Col B A M) by (forward_using lemma_collinearorder).

assert (Col B D m) by (forward_using lemma_collinearorder).

assert (Par E B C A) by (conclude_def PG ).

assert (nCol E B A) by (forward_using lemma_parallelNC).

assert (nCol B A E) by (forward_using lemma_NCorder).

assert (TS E B A C) by (conclude_def TS ).

assert (TS C B A E) by (conclude lemma_oppositesidesymmetric).

assert (Par D F B C) by (conclude_def PG ).

assert (nCol D F B) by (forward_using lemma_parallelNC).

assert (nCol B D F) by (forward_using lemma_NCorder).

assert (TS F B D C) by (conclude_def TS ).

assert (TS C B D F) by (conclude lemma_oppositesidesymmetric).

assert (ET B E A A C B) by (conclude axiom_congruentequal).

assert (ET B E A C B A) by (forward_using axiom_ETpermutation).

assert (ET C B A B E A) by (conclude axiom_ETsymmetric).

assert (ET C B A B A E) by (forward_using axiom_ETpermutation).

assert (ET B F D D C B) by (conclude axiom_congruentequal).

assert (ET B F D C B D) by (forward_using axiom_ETpermutation).

assert (ET C B D B F D) by (conclude axiom_ETsymmetric).

assert (ET C B D B D F) by (forward_using axiom_ETpermutation).

assert (EF E B C A C B F D) by (forward_using axiom_EFpermutation).

assert (EF C B F D E B C A) by (conclude axiom_EFsymmetric).

assert (EF C B F D C B E A) by (forward_using axiom_EFpermutation).

assert (EF C B E A C B F D) by (conclude axiom_EFsymmetric).

assert (ET C B A C B D) by (conclude axiom_halvesofequals).

assert (ET C B A D B C) by (forward_using axiom_ETpermutation).

assert (ET D B C C B A) by (conclude axiom_ETsymmetric).

assert (ET D B C A B C) by (forward_using axiom_ETpermutation).

assert (ET A B C D B C) by (conclude axiom_ETsymmetric).

close.

Qed.

End Euclid.

Require Export GeoCoq.Elements.OriginalProofs.lemma_PGrotate.

Require Export GeoCoq.Elements.OriginalProofs.proposition_35.

Section Euclid.

Context `{Ax:area}.

Lemma proposition_37 :

∀ A B C D,

Par A D B C →

ET A B C D B C.

Proof.

intros.

assert (Par B C A D) by (conclude lemma_parallelsymmetric).

assert (Par C B A D) by (forward_using lemma_parallelflip).

assert (eq A A) by (conclude cn_equalityreflexive).

assert (eq D D) by (conclude cn_equalityreflexive).

assert (Col A D A) by (conclude_def Col ).

assert (Col A D D) by (conclude_def Col ).

let Tf:=fresh in

assert (Tf:∃ E, (PG A E B C ∧ Col A D E)) by (conclude lemma_triangletoparallelogram);destruct Tf as [E];spliter.

let Tf:=fresh in

assert (Tf:∃ F, (PG D F B C ∧ Col A D F)) by (conclude lemma_triangletoparallelogram);destruct Tf as [F];spliter.

assert (PG E B C A) by (conclude lemma_PGrotate).

assert (PG F B C D) by (conclude lemma_PGrotate).

assert (Col D A F) by (forward_using lemma_collinearorder).

assert (Col D A E) by (forward_using lemma_collinearorder).

assert (nCol C A D) by (forward_using lemma_parallelNC).

assert (neq A D) by (forward_using lemma_NCdistinct).

assert (neq D A) by (conclude lemma_inequalitysymmetric).

assert (Col A F E) by (conclude lemma_collinear4).

assert (Col E A D) by (forward_using lemma_collinearorder).

assert (Col E A F) by (forward_using lemma_collinearorder).

assert (EF E B C A F B C D) by (conclude proposition_35).

assert (Cong_3 B E A A C B) by (conclude proposition_34).

assert (Cong_3 B F D D C B) by (conclude proposition_34).

let Tf:=fresh in

assert (Tf:∃ M, (BetS E M C ∧ BetS B M A)) by (conclude lemma_diagonalsmeet);destruct Tf as [M];spliter.

let Tf:=fresh in

assert (Tf:∃ m, (BetS F m C ∧ BetS B m D)) by (conclude lemma_diagonalsmeet);destruct Tf as [m];spliter.

assert (Col B M A) by (conclude_def Col ).

assert (Col B m D) by (conclude_def Col ).

assert (Col B A M) by (forward_using lemma_collinearorder).

assert (Col B D m) by (forward_using lemma_collinearorder).

assert (Par E B C A) by (conclude_def PG ).

assert (nCol E B A) by (forward_using lemma_parallelNC).

assert (nCol B A E) by (forward_using lemma_NCorder).

assert (TS E B A C) by (conclude_def TS ).

assert (TS C B A E) by (conclude lemma_oppositesidesymmetric).

assert (Par D F B C) by (conclude_def PG ).

assert (nCol D F B) by (forward_using lemma_parallelNC).

assert (nCol B D F) by (forward_using lemma_NCorder).

assert (TS F B D C) by (conclude_def TS ).

assert (TS C B D F) by (conclude lemma_oppositesidesymmetric).

assert (ET B E A A C B) by (conclude axiom_congruentequal).

assert (ET B E A C B A) by (forward_using axiom_ETpermutation).

assert (ET C B A B E A) by (conclude axiom_ETsymmetric).

assert (ET C B A B A E) by (forward_using axiom_ETpermutation).

assert (ET B F D D C B) by (conclude axiom_congruentequal).

assert (ET B F D C B D) by (forward_using axiom_ETpermutation).

assert (ET C B D B F D) by (conclude axiom_ETsymmetric).

assert (ET C B D B D F) by (forward_using axiom_ETpermutation).

assert (EF E B C A C B F D) by (forward_using axiom_EFpermutation).

assert (EF C B F D E B C A) by (conclude axiom_EFsymmetric).

assert (EF C B F D C B E A) by (forward_using axiom_EFpermutation).

assert (EF C B E A C B F D) by (conclude axiom_EFsymmetric).

assert (ET C B A C B D) by (conclude axiom_halvesofequals).

assert (ET C B A D B C) by (forward_using axiom_ETpermutation).

assert (ET D B C C B A) by (conclude axiom_ETsymmetric).

assert (ET D B C A B C) by (forward_using axiom_ETpermutation).

assert (ET A B C D B C) by (conclude axiom_ETsymmetric).

close.

Qed.

End Euclid.