Proof with eTarski.
intros.
assert (Bet B C C)...
assert (exists x, Bet B x B /\ Bet C x A)...
DecompExAnd H1 x.
assert (B=x)...
congruence.
Qed.
Hint Resolve between_symmetry : sTarski.

Lemma beetween_trivial2 : forall A B : Point, Bet A A B.
Hint Resolve beetween_trivial beetween_trivial2 : Tarski.