Ltac let_symmetric C P A :=
let id1:=fresh in (assert (id1:(exists A', Midpoint P A A'));
[apply symmetric_point_construction|ex_and id1 C]).Ltac symmetric B' A B :=
assert(sp:= symmetric_point_construction B A); ex_and sp B'.

Lemma perp_in_dec : forall X A B C D, Perp_at X A B C D \/ ~ Perp_at X A B C D.