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_10 : forall A B C A' B' C', Per A B C -> Cong_3 A B C A' B' C' -> Per A' B' C'.