Require Export GeoCoq.Elements.OriginalProofs.lemma_NChelper.
Require Export GeoCoq.Elements.OriginalProofs.lemma_pointreflectionisometry.
Require Export GeoCoq.Elements.OriginalProofs.lemma_oppositesidesymmetric.
Require Export GeoCoq.Elements.OriginalProofs.proposition_27.
Require Export GeoCoq.Elements.OriginalProofs.lemma_parallelflip.
Section Euclid.
Context `{Ax:euclidean_neutral_ruler_compass}.

Lemma proposition_31 :
forall A B C D,
BetS B D C -> nCol B C A ->
exists X Y Z, BetS X A Y /\ CongA Y A D A D B /\ CongA Y A D B D A /\ CongA D A Y B D A /\ CongA X A D A D C /\ CongA X A D C D A /\ CongA D A X C D A /\ Par X Y B C /\ Cong X A D C /\ Cong A Y B D /\ Cong A Z Z D /\ Cong X Z Z C /\ Cong B Z Z Y /\ BetS X Z C /\ BetS B Z Y /\ BetS A Z D.