Require Export GeoCoq.Elements.OriginalProofs.proposition_44.
Require Export GeoCoq.Elements.OriginalProofs.lemma_RTcongruence.
Require Export GeoCoq.Elements.OriginalProofs.lemma_RTsymmetric.
Require Export GeoCoq.Elements.OriginalProofs.proposition_14.
Section Euclid.
Context `{Ax:area}.

Lemma proposition_45 :
forall A B C D E J K N O R S,
nCol J E N -> nCol A B D -> nCol C B D -> BetS A O C -> BetS B O D -> neq R K -> nCol K R S ->
exists X Z U, PG X K Z U /\ CongA X K Z J E N /\ EF X K Z U A B C D /\ Out K R Z /\ OS X S K Z.