Ltac let_symmetric C P A := 
let id1:=fresh in (assert (id1:(exists A', is_midpoint P A A'));
[apply symmetric_point_construction|DecompEx id1 C]).

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'.