proposition: Lemma proposition_47 : forall A B C D E F G H K, Triangle A B C -> Per B A C -> SQ A B F G -> TS G B A C -> SQ A C K H -> TS H C A B -> SQ B C E D -> TS D C B A -> exists X Y, PG B X Y D /\ BetS B X C /\ PG X C E Y /\ BetS D Y E /\ EF A B F G B X Y D /\ EF A C K H X C E Y.
proof: Proof. intros. let Tf:=fresh in assert (Tf:exists M L, (PG B M L D /\ BetS B M C /\ PG M C E L /\ BetS D L E /\ BetS L M A /\ Per D L A /\ EF A B F G B M L D)) by (conclude proposition_47B);destruct Tf as [M[L]];spliter. assert (nCol A B C) by (conclude_def Triangle ). assert (nCol A C B) by (forward_using lemma_NCorder). assert (Triangle A C B) by (conclude_def Triangle ). assert (Per C A B) by (conclude lemma_8_2). assert (SQ C B D E) by (conclude lemma_squareflip). assert (TS D B C A) by (conclude lemma_oppositesideflip). assert (PG B C E D) by (conclude lemma_squareparallelogram). assert (Par B C E D) by (conclude_def PG ). assert (TP B C E D) by (conclude lemma_paralleldef2B). assert (OS E D B C) by (conclude_def TP ). assert (TS E B C A) by (conclude lemma_planeseparation). let Tf:=fresh in assert (Tf:exists m l, (PG C m l E /\ BetS C m B /\ PG m B D l /\ BetS E l D /\ BetS l m A /\ Per E l A /\ EF A C K H C m l E)) by (conclude proposition_47B);destruct Tf as [m[l]];spliter. assert (neq E D) by (forward_using lemma_betweennotequal). assert (neq D E) by (conclude lemma_inequalitysymmetric). assert (Col E l D) by (conclude_def Col ). assert (Col D L E) by (conclude_def Col ). assert (Col D E L) by (forward_using lemma_collinearorder). assert (Col D E l) by (forward_using lemma_collinearorder). assert (Col E l L) by (conclude lemma_collinear4). assert (neq L E) by (forward_using lemma_betweennotequal). assert (neq E L) by (conclude lemma_inequalitysymmetric). assert (Per E L A) by (conclude lemma_collinearright). assert (eq l L) by (conclude lemma_droppedperpendicularunique). assert (eq L l) by (conclude lemma_equalitysymmetric). assert (BetS L m A) by (conclude cn_equalitysub). assert (BetS C M B) by (conclude axiom_betweennesssymmetry). assert (Col L m A) by (conclude_def Col ). assert (Col L M A) by (conclude_def Col ). assert (Col L A m) by (forward_using lemma_collinearorder). assert (Col L A M) by (forward_using lemma_collinearorder). assert (neq L A) by (forward_using lemma_betweennotequal). assert (Col A m M) by (conclude lemma_collinear4). assert (Col M m A) by (forward_using lemma_collinearorder). assert (Col B M C) by (conclude_def Col ). assert (Col C B M) by (forward_using lemma_collinearorder). assert (Col C m B) by (conclude_def Col ). assert (Col C B m) by (forward_using lemma_collinearorder). assert (neq C B) by (forward_using lemma_betweennotequal). assert (Col B M m) by (conclude lemma_collinear4). assert (Col M m B) by (forward_using lemma_collinearorder). assert (~ neq M m). { intro. assert (Col m M A) by (forward_using lemma_collinearorder). assert (Col m M B) by (forward_using lemma_collinearorder). assert (neq m M) by (conclude lemma_inequalitysymmetric). assert (Col M A B) by (conclude lemma_collinear4). assert (Col M B A) by (forward_using lemma_collinearorder). assert (Col M B C) by (forward_using lemma_collinearorder). assert (neq M B) by (forward_using lemma_betweennotequal). assert (Col B A C) by (conclude lemma_collinear4). assert (Col A B C) by (forward_using lemma_collinearorder). contradict. } assert (EF A C K H C M l E) by (conclude cn_equalitysub). assert (EF A C K H C M L E) by (conclude cn_equalitysub). assert (EF A C K H M C E L) by (forward_using axiom_EFpermutation). close. Qed.
imports: GeoCoq.Elements.OriginalProofs.proposition_47B GeoCoq.Elements.OriginalProofs.lemma_squareflip