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 per_perp_in : forall A B C, A <> B -> B <> C -> Per A B C -> Perp_at B A B B C.