Definition contraction a b k := k *: (b - a) + a.

Lemma euclid a b c d t (k1 := betR a d t) (k2 := betR b d c) :
bet a d t -> bet b d c -> b <> d -> d <> c ->
~ (bet a b c \/ bet b c a \/ bet c a b) ->
exists x y, bet a b x /\ bet a c y /\ bet x t y.