proposition: Lemma lemma_notperp : forall A B C P, BetS A C B -> nCol A B P -> exists X, nCol A B X /\ OS X P A B /\ ~ Per A C X.
proof: Proof. intros. assert (neq C B) by (forward_using lemma_betweennotequal). assert (neq B C) by (conclude lemma_inequalitysymmetric). assert (~ eq C P). { intro. assert (nCol A B C) by (conclude cn_equalitysub). assert (Col A C B) by (conclude_def Col ). assert (Col A B C) by (forward_using lemma_collinearorder). contradict. } let Tf:=fresh in assert (Tf:exists Q, (BetS B C Q /\ Cong C Q C P)) by (conclude lemma_extension);destruct Tf as [Q];spliter. assert (~ eq P Q). { intro. assert (Col B C Q) by (conclude_def Col ). assert (Col B C P) by (conclude cn_equalitysub). assert (Col A C B) by (conclude_def Col ). assert (Col C B A) by (forward_using lemma_collinearorder). assert (Col C B P) by (forward_using lemma_collinearorder). assert (Col B A P) by (conclude lemma_collinear4). assert (Col A B P) by (forward_using lemma_collinearorder). contradict. } let Tf:=fresh in assert (Tf:exists M, (BetS P M Q /\ Cong M P M Q)) by (conclude proposition_10);destruct Tf as [M];spliter. assert (Col A C B) by (conclude_def Col ). assert (Col C B A) by (forward_using lemma_collinearorder). assert (neq C B) by (forward_using lemma_betweennotequal). assert (Col C B Q) by (conclude_def Col ). assert (Col B A Q) by (conclude lemma_collinear4). assert (Col A B Q) by (forward_using lemma_collinearorder). assert (neq A B) by (forward_using lemma_betweennotequal). assert (OS P P A B) by (conclude lemma_samesidereflexive). assert (neq Q P) by (conclude lemma_inequalitysymmetric). assert (BetS Q M P) by (conclude axiom_betweennesssymmetry). assert (Out Q P M) by (conclude lemma_ray4). assert (Col A Q B) by (forward_using lemma_collinearorder). assert (OS P M A B) by (conclude lemma_sameside2). assert (OS M P A B) by (forward_using lemma_samesidesymmetric). assert (~ Col A B M). { intro. assert (Col B M Q) by (conclude lemma_collinear4). assert (Col P M Q) by (conclude_def Col ). assert (Col M Q P) by (forward_using lemma_collinearorder). assert (Col M Q B) by (forward_using lemma_collinearorder). assert (neq M Q) by (forward_using lemma_betweennotequal). assert (Col Q P B) by (conclude lemma_collinear4). assert (Col Q B P) by (forward_using lemma_collinearorder). assert (Col Q B A) by (forward_using lemma_collinearorder). assert (neq B Q) by (forward_using lemma_betweennotequal). assert (neq Q B) by (conclude lemma_inequalitysymmetric). assert (Col B P A) by (conclude lemma_collinear4). assert (Col A B P) by (forward_using lemma_collinearorder). contradict. } let Tf:=fresh in assert (Tf:exists R, Perp_at M R A B R) by (conclude proposition_12);destruct Tf as [R];spliter. let Tf:=fresh in assert (Tf:exists E, (Col M R R /\ Col A B R /\ Col A B E /\ Per E R M)) by (conclude_def Perp_at );destruct Tf as [E];spliter. assert (Per M R E) by (conclude lemma_8_2). assert (~ eq M R). { intro. assert (Col A B M) by (conclude cn_equalitysub). contradict. } assert (neq A C) by (forward_using lemma_betweennotequal). assert (Col A B C) by (forward_using lemma_collinearorder). assert (~ Per A C M). { intro. assert (~ neq R C). { intro. assert (Col B A C) by (forward_using lemma_collinearorder). assert (Col B A R) by (forward_using lemma_collinearorder). assert (neq B A) by (conclude lemma_inequalitysymmetric). assert (Col A C R) by (conclude lemma_collinear4). assert (Per R C M) by (conclude lemma_collinearright). assert (~ Per M R C) by (conclude lemma_8_7). assert (Per E R M) by (conclude lemma_8_2). assert (Col B C R) by (conclude lemma_collinear4). assert (Col B C E) by (conclude lemma_collinear4). assert (neq C B) by (forward_using lemma_betweennotequal). assert (neq B C) by (conclude lemma_inequalitysymmetric). assert (Col C R E) by (conclude lemma_collinear4). assert (Col E R C) by (forward_using lemma_collinearorder). assert (neq C R) by (conclude lemma_inequalitysymmetric). assert (Per C R M) by (conclude lemma_collinearright). assert (Per M R C) by (conclude lemma_8_2). contradict. } assert (~ eq M C). { intro. assert (Col A B M) by (conclude cn_equalitysub). contradict. } assert (Cong Q C P C) by (forward_using lemma_congruenceflip). assert (BetS Q M P) by (conclude axiom_betweennesssymmetry). assert (Cong Q M P M) by (forward_using lemma_doublereverse). assert (Per Q M C) by (conclude_def Per ). assert (neq C Q) by (forward_using lemma_betweennotequal). assert (neq Q C) by (conclude lemma_inequalitysymmetric). assert (Per E R M) by (conclude lemma_8_2). assert (neq Q R) by (conclude cn_equalitysub). assert (Col B C Q) by (conclude_def Col ). assert (Col C B Q) by (forward_using lemma_collinearorder). assert (Col Q B C) by (forward_using lemma_collinearorder). assert (Col B E R) by (conclude lemma_collinear4). assert (Col B E Q) by (conclude lemma_collinear4). assert (~ neq B E). { intro. assert (Col E R Q) by (conclude lemma_collinear4). assert (Per Q R M) by (conclude lemma_collinearright). assert (Per Q C M) by (conclude cn_equalitysub). assert (Per M C Q) by (conclude lemma_8_2). assert (~ Per Q M C) by (conclude lemma_8_7). contradict. } assert (Col A E R) by (conclude cn_equalitysub). assert (Col A B Q) by (forward_using lemma_collinearorder). assert (Col A E Q) by (conclude cn_equalitysub). assert (neq A E) by (conclude cn_equalitysub). assert (Col E R Q) by (conclude lemma_collinear4). assert (Per Q R M) by (conclude lemma_collinearright). assert (Per Q C M) by (conclude cn_equalitysub). assert (Per M C Q) by (conclude lemma_8_2). assert (~ Per Q M C) by (conclude lemma_8_7). contradict. } exists M;close. Qed.
imports: GeoCoq.Elements.OriginalProofs.lemma_samesidereflexive GeoCoq.Elements.OriginalProofs.lemma_sameside2 GeoCoq.Elements.OriginalProofs.lemma_samesidesymmetric GeoCoq.Elements.OriginalProofs.proposition_12 GeoCoq.Elements.OriginalProofs.lemma_8_7