# Library GeoCoq.Elements.OriginalProofs.lemma_diagonalsbisect

Require Export GeoCoq.Elements.OriginalProofs.lemma_parallelNC.

Require Export GeoCoq.Elements.OriginalProofs.lemma_crossimpliesopposite.

Require Export GeoCoq.Elements.OriginalProofs.proposition_34.

Require Export GeoCoq.Elements.OriginalProofs.lemma_NCdistinct.

Section Euclid.

Context `{Ax:euclidean_euclidean}.

Lemma lemma_diagonalsbisect :

∀ A B C D,

PG A B C D →

∃ X, Midpoint A X C ∧ Midpoint B X D.

Proof.

intros.

let Tf:=fresh in

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

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

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

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

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

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

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

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

assert (TS A B D C) by (forward_using lemma_crossimpliesopposite).

assert (CongA A B D B D C) by (conclude proposition_29B).

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

assert (BetS C M A) by (conclude axiom_betweennesssymmetry).

assert (BetS D M B) by (conclude axiom_betweennesssymmetry).

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

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

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

assert (TS B A C D) by (forward_using lemma_crossimpliesopposite).

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

assert (CongA B A C A C D) by (conclude proposition_29B).

assert (Cong A B D C) by (forward_using proposition_34).

assert (Cong A B C D) by (forward_using lemma_congruenceflip).

assert (¬ Col M A B).

{

intro.

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

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

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

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

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

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

contradict.

}

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

assert (¬ Col M C D).

{

intro.

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

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

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

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

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

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

contradict.

}

assert (Triangle M C D) by (conclude_def Triangle ).

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

assert (CongA B A C A C D) by (conclude proposition_29B).

assert (CongA B A C B A C) by (conclude lemma_equalanglesreflexive).

assert (Out A C M) by (conclude lemma_ray4).

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

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

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

assert (Out A B B) by (conclude lemma_ray4).

assert (CongA B A C B A M) by (conclude lemma_equalangleshelper).

assert (CongA B A M B A C) by (conclude lemma_equalanglessymmetric).

assert (CongA B A M A C D) by (conclude lemma_equalanglestransitive).

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

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

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

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

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

assert (Out C D D) by (conclude lemma_ray4).

assert (Out C A M) by (conclude lemma_ray4).

assert (CongA A C D A C D) by (conclude lemma_equalanglesreflexive).

assert (CongA A C D M C D) by (conclude lemma_equalangleshelper).

assert (CongA B A M M C D) by (conclude lemma_equalanglestransitive).

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

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

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

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

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

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

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

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

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

assert (nCol M C D) by (conclude lemma_NChelper).

assert (CongA M C D D C M) by (conclude lemma_ABCequalsCBA).

assert (CongA B A M D C M) by (conclude lemma_equalanglestransitive).

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

assert (CongA A B D B D C) by (conclude proposition_29B).

assert (CongA A B D A B D) by (conclude lemma_equalanglesreflexive).

assert (Out B D M) by (conclude lemma_ray4).

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

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

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

assert (Out B A A) by (conclude lemma_ray4).

assert (CongA A B D A B M) by (conclude lemma_equalangleshelper).

assert (CongA A B M A B D) by (conclude lemma_equalanglessymmetric).

assert (CongA A B M B D C) by (conclude lemma_equalanglestransitive).

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

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

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

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

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

assert (Out D C C) by (conclude lemma_ray4).

assert (Out D B M) by (conclude lemma_ray4).

assert (CongA B D C B D C) by (conclude lemma_equalanglesreflexive).

assert (CongA B D C M D C) by (conclude lemma_equalangleshelper).

assert (CongA A B M M D C) by (conclude lemma_equalanglestransitive).

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

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

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

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

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

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

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

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

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

assert (nCol M D C) by (conclude lemma_NChelper).

assert (CongA M D C C D M) by (conclude lemma_ABCequalsCBA).

assert (CongA A B M C D M) by (conclude lemma_equalanglestransitive).

assert (CongA M A B M C D) by (conclude lemma_equalanglesflip).

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

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

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

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

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

assert (nCol A M B) by (conclude lemma_NChelper).

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

assert (nCol M D C) by (forward_using lemma_NCorder).

assert ((Cong M A M C ∧ Cong M B M D ∧ CongA A M B C M D)) by (conclude proposition_26A).

assert (Cong A M M C) by (forward_using lemma_congruenceflip).

assert (Cong B M M D) by (forward_using lemma_congruenceflip).

assert (Midpoint A M C) by (conclude_def Midpoint ).

assert (Midpoint B M D) by (conclude_def Midpoint ).

close.

Qed.

End Euclid.

Require Export GeoCoq.Elements.OriginalProofs.lemma_crossimpliesopposite.

Require Export GeoCoq.Elements.OriginalProofs.proposition_34.

Require Export GeoCoq.Elements.OriginalProofs.lemma_NCdistinct.

Section Euclid.

Context `{Ax:euclidean_euclidean}.

Lemma lemma_diagonalsbisect :

∀ A B C D,

PG A B C D →

∃ X, Midpoint A X C ∧ Midpoint B X D.

Proof.

intros.

let Tf:=fresh in

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

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

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

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

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

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

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

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

assert (TS A B D C) by (forward_using lemma_crossimpliesopposite).

assert (CongA A B D B D C) by (conclude proposition_29B).

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

assert (BetS C M A) by (conclude axiom_betweennesssymmetry).

assert (BetS D M B) by (conclude axiom_betweennesssymmetry).

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

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

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

assert (TS B A C D) by (forward_using lemma_crossimpliesopposite).

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

assert (CongA B A C A C D) by (conclude proposition_29B).

assert (Cong A B D C) by (forward_using proposition_34).

assert (Cong A B C D) by (forward_using lemma_congruenceflip).

assert (¬ Col M A B).

{

intro.

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

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

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

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

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

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

contradict.

}

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

assert (¬ Col M C D).

{

intro.

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

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

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

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

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

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

contradict.

}

assert (Triangle M C D) by (conclude_def Triangle ).

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

assert (CongA B A C A C D) by (conclude proposition_29B).

assert (CongA B A C B A C) by (conclude lemma_equalanglesreflexive).

assert (Out A C M) by (conclude lemma_ray4).

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

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

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

assert (Out A B B) by (conclude lemma_ray4).

assert (CongA B A C B A M) by (conclude lemma_equalangleshelper).

assert (CongA B A M B A C) by (conclude lemma_equalanglessymmetric).

assert (CongA B A M A C D) by (conclude lemma_equalanglestransitive).

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

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

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

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

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

assert (Out C D D) by (conclude lemma_ray4).

assert (Out C A M) by (conclude lemma_ray4).

assert (CongA A C D A C D) by (conclude lemma_equalanglesreflexive).

assert (CongA A C D M C D) by (conclude lemma_equalangleshelper).

assert (CongA B A M M C D) by (conclude lemma_equalanglestransitive).

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

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

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

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

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

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

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

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

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

assert (nCol M C D) by (conclude lemma_NChelper).

assert (CongA M C D D C M) by (conclude lemma_ABCequalsCBA).

assert (CongA B A M D C M) by (conclude lemma_equalanglestransitive).

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

assert (CongA A B D B D C) by (conclude proposition_29B).

assert (CongA A B D A B D) by (conclude lemma_equalanglesreflexive).

assert (Out B D M) by (conclude lemma_ray4).

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

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

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

assert (Out B A A) by (conclude lemma_ray4).

assert (CongA A B D A B M) by (conclude lemma_equalangleshelper).

assert (CongA A B M A B D) by (conclude lemma_equalanglessymmetric).

assert (CongA A B M B D C) by (conclude lemma_equalanglestransitive).

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

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

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

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

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

assert (Out D C C) by (conclude lemma_ray4).

assert (Out D B M) by (conclude lemma_ray4).

assert (CongA B D C B D C) by (conclude lemma_equalanglesreflexive).

assert (CongA B D C M D C) by (conclude lemma_equalangleshelper).

assert (CongA A B M M D C) by (conclude lemma_equalanglestransitive).

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

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

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

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

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

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

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

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

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

assert (nCol M D C) by (conclude lemma_NChelper).

assert (CongA M D C C D M) by (conclude lemma_ABCequalsCBA).

assert (CongA A B M C D M) by (conclude lemma_equalanglestransitive).

assert (CongA M A B M C D) by (conclude lemma_equalanglesflip).

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

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

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

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

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

assert (nCol A M B) by (conclude lemma_NChelper).

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

assert (nCol M D C) by (forward_using lemma_NCorder).

assert ((Cong M A M C ∧ Cong M B M D ∧ CongA A M B C M D)) by (conclude proposition_26A).

assert (Cong A M M C) by (forward_using lemma_congruenceflip).

assert (Cong B M M D) by (forward_using lemma_congruenceflip).

assert (Midpoint A M C) by (conclude_def Midpoint ).

assert (Midpoint B M D) by (conclude_def Midpoint ).

close.

Qed.

End Euclid.