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 l8_15_1 : forall A B C X, Col A B X -> Perp A B C X -> Perp_at X A B C X.