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]).

(* Specifications: *)
Lemma l8_7 : forall A B C, Per A B C -> Per A C B -> B=C.
Lemma l8_8 : forall A B, Per A B A -> A=B.
Lemma l8_9 : forall A B C, Per A B C -> Col A B C -> A=B \/ C=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'.