proposition: Lemma proposition_02 : forall A B C, neq A B -> neq B C -> exists X, Cong A X B C.
proof: Proof. intros. let Tf:=fresh in assert (Tf:exists D, (equilateral A B D /\ Triangle A B D)) by (conclude proposition_01);destruct Tf as [D];spliter. assert (Cong A B B D) by (conclude_def equilateral ). assert (Cong B D A B) by (conclude lemma_congruencesymmetric). assert (Cong B D B A) by (forward_using lemma_congruenceflip). assert (Cong B D D A) by (conclude_def equilateral ). assert (nCol A B D) by (conclude_def Triangle ). assert (eq B B) by (conclude cn_equalityreflexive). let Tf:=fresh in assert (Tf:exists J, CI J B B C) by (conclude postulate_Euclid3);destruct Tf as [J];spliter. assert (InCirc B J) by (conclude_def InCirc ). assert (neq D B) by (forward_using lemma_NCdistinct). let Tf:=fresh in assert (Tf:exists P G, (Col D B P /\ BetS D B G /\ OnCirc P J /\ OnCirc G J /\ BetS P B G)) by (conclude postulate_line_circle);destruct Tf as [P[G]];spliter. assert (neq D G) by (forward_using lemma_betweennotequal). let Tf:=fresh in assert (Tf:exists R, CI R D D G) by (conclude postulate_Euclid3);destruct Tf as [R];spliter. assert (Cong B G B C) by (conclude axiom_circle_center_radius). assert (Cong B C B G) by (conclude lemma_congruencesymmetric). assert (eq D D) by (conclude cn_equalityreflexive). assert (Cong D A B D) by (conclude lemma_congruencesymmetric). assert (Cong D A D B) by (forward_using lemma_congruenceflip). assert (Cong D G D G) by (conclude cn_congruencereflexive). assert (InCirc A R) by (conclude_def InCirc ). assert (neq D A) by (forward_using lemma_NCdistinct). let Tf:=fresh in assert (Tf:exists Q L, (Col D A Q /\ BetS D A L /\ OnCirc Q R /\ OnCirc L R /\ BetS Q A L)) by (conclude postulate_line_circle);destruct Tf as [Q[L]];spliter. assert (Cong D L D G) by (conclude axiom_circle_center_radius). assert (Cong D B D B) by (conclude cn_congruencereflexive). assert (Cong D B D A) by (conclude lemma_congruencesymmetric). assert (Cong D G D L) by (conclude lemma_congruencesymmetric). assert (Cong B G A L) by (conclude lemma_differenceofparts). assert (Cong A L B C) by (conclude cn_congruencetransitive). close. Unshelve. exact A. exact A. Qed.
imports: GeoCoq.Elements.OriginalProofs.proposition_01 GeoCoq.Elements.OriginalProofs.lemma_congruenceflip GeoCoq.Elements.OriginalProofs.lemma_NCdistinct GeoCoq.Elements.OriginalProofs.lemma_differenceofparts