# Library GeoCoq.Elements.OriginalProofs.proposition_29B

Require Export GeoCoq.Elements.OriginalProofs.proposition_29.

Section Euclid.

Context `{Ax:euclidean_euclidean}.

Lemma proposition_29B :

∀ A D G H,

Par A G H D → TS A G H D →

CongA A G H G H D.

Proof.

intros.

let Tf:=fresh in

assert (Tf:∃ a d g h m, (neq A G ∧ neq H D ∧ Col A G a ∧ Col A G g ∧ neq a g ∧ Col H D h ∧ Col H D d ∧ neq h d ∧ ¬ Meet A G H D ∧ BetS a m d ∧ BetS h m g)) by (conclude_def Par );destruct Tf as [a[d[g[h[m]]]]];spliter.

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

assert (¬ eq H G).

{

intro.

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

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

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

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

assert (Col A G H) by (conclude cn_equalitysub).

assert (Meet A G H D) by (conclude_def Meet ).

contradict.

}

assert (neq G H) by (conclude lemma_inequalitysymmetric).

let Tf:=fresh in

assert (Tf:∃ F, (BetS G H F ∧ Cong H F G H)) by (conclude postulate_extension);destruct Tf as [F];spliter.

let Tf:=fresh in

assert (Tf:∃ B, (BetS A G B ∧ Cong G B A G)) by (conclude postulate_extension);destruct Tf as [B];spliter.

let Tf:=fresh in

assert (Tf:∃ C, (BetS D H C ∧ Cong H C D H)) by (conclude postulate_extension);destruct Tf as [C];spliter.

let Tf:=fresh in

assert (Tf:∃ E, (BetS H G E ∧ Cong G E H G)) by (conclude postulate_extension);destruct Tf as [E];spliter.

assert (neq A B) by (forward_using lemma_betweennotequal).

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

assert (neq D C) by (forward_using lemma_betweennotequal).

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

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

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

assert (Col G A a) by (forward_using lemma_collinearorder).

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

assert (Col A B a) by (conclude lemma_collinear4).

assert (Col G A g) by (forward_using lemma_collinearorder).

assert (Col A B g) by (conclude lemma_collinear4).

assert (Col D H C) by (conclude_def Col ).

assert (Col H D C) by (forward_using lemma_collinearorder).

assert (Col D C h) by (conclude lemma_collinear4).

assert (Col C D h) by (forward_using lemma_collinearorder).

assert (Col D d C) by (conclude lemma_collinear4).

assert (Col C D d) by (forward_using lemma_collinearorder).

assert (¬ Meet A B C D).

{

intro.

let Tf:=fresh in

assert (Tf:∃ M, (neq A B ∧ neq C D ∧ Col A B M ∧ Col C D M)) by (conclude_def Meet );destruct Tf as [M];spliter.

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

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

assert (Col A G M) by (conclude lemma_collinear4).

assert (Col C D H) by (forward_using lemma_collinearorder).

assert (Col D H M) by (conclude lemma_collinear4).

assert (Col H D M) by (forward_using lemma_collinearorder).

assert (Meet A G H D) by (conclude_def Meet ).

contradict.

}

assert (Par A B C D) by (conclude_def Par ).

assert (BetS C H D) by (conclude axiom_betweennesssymmetry).

assert (BetS E G H) by (conclude axiom_betweennesssymmetry).

assert ((CongA A G H G H D ∧ CongA E G B G H D ∧ RT B G H G H D)) by (conclude proposition_29).

close.

Qed.

End Euclid.

Section Euclid.

Context `{Ax:euclidean_euclidean}.

Lemma proposition_29B :

∀ A D G H,

Par A G H D → TS A G H D →

CongA A G H G H D.

Proof.

intros.

let Tf:=fresh in

assert (Tf:∃ a d g h m, (neq A G ∧ neq H D ∧ Col A G a ∧ Col A G g ∧ neq a g ∧ Col H D h ∧ Col H D d ∧ neq h d ∧ ¬ Meet A G H D ∧ BetS a m d ∧ BetS h m g)) by (conclude_def Par );destruct Tf as [a[d[g[h[m]]]]];spliter.

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

assert (¬ eq H G).

{

intro.

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

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

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

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

assert (Col A G H) by (conclude cn_equalitysub).

assert (Meet A G H D) by (conclude_def Meet ).

contradict.

}

assert (neq G H) by (conclude lemma_inequalitysymmetric).

let Tf:=fresh in

assert (Tf:∃ F, (BetS G H F ∧ Cong H F G H)) by (conclude postulate_extension);destruct Tf as [F];spliter.

let Tf:=fresh in

assert (Tf:∃ B, (BetS A G B ∧ Cong G B A G)) by (conclude postulate_extension);destruct Tf as [B];spliter.

let Tf:=fresh in

assert (Tf:∃ C, (BetS D H C ∧ Cong H C D H)) by (conclude postulate_extension);destruct Tf as [C];spliter.

let Tf:=fresh in

assert (Tf:∃ E, (BetS H G E ∧ Cong G E H G)) by (conclude postulate_extension);destruct Tf as [E];spliter.

assert (neq A B) by (forward_using lemma_betweennotequal).

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

assert (neq D C) by (forward_using lemma_betweennotequal).

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

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

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

assert (Col G A a) by (forward_using lemma_collinearorder).

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

assert (Col A B a) by (conclude lemma_collinear4).

assert (Col G A g) by (forward_using lemma_collinearorder).

assert (Col A B g) by (conclude lemma_collinear4).

assert (Col D H C) by (conclude_def Col ).

assert (Col H D C) by (forward_using lemma_collinearorder).

assert (Col D C h) by (conclude lemma_collinear4).

assert (Col C D h) by (forward_using lemma_collinearorder).

assert (Col D d C) by (conclude lemma_collinear4).

assert (Col C D d) by (forward_using lemma_collinearorder).

assert (¬ Meet A B C D).

{

intro.

let Tf:=fresh in

assert (Tf:∃ M, (neq A B ∧ neq C D ∧ Col A B M ∧ Col C D M)) by (conclude_def Meet );destruct Tf as [M];spliter.

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

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

assert (Col A G M) by (conclude lemma_collinear4).

assert (Col C D H) by (forward_using lemma_collinearorder).

assert (Col D H M) by (conclude lemma_collinear4).

assert (Col H D M) by (forward_using lemma_collinearorder).

assert (Meet A G H D) by (conclude_def Meet ).

contradict.

}

assert (Par A B C D) by (conclude_def Par ).

assert (BetS C H D) by (conclude axiom_betweennesssymmetry).

assert (BetS E G H) by (conclude axiom_betweennesssymmetry).

assert ((CongA A G H G H D ∧ CongA E G B G H D ∧ RT B G H G H D)) by (conclude proposition_29).

close.

Qed.

End Euclid.