proposition: Lemma lemma_diagonalsbisect : forall A B C D, PG A B C D -> exists X, Midpoint A X C /\ Midpoint B X D.
proof: Proof. intros. let Tf:=fresh in assert (Tf:exists 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 (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 (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 (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 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 (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 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 ((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.
imports: GeoCoq.Elements.OriginalProofs.lemma_parallelNC GeoCoq.Elements.OriginalProofs.lemma_crossimpliesopposite GeoCoq.Elements.OriginalProofs.proposition_34 GeoCoq.Elements.OriginalProofs.lemma_NCdistinct