proposition: Lemma extensionP a b c : a != b -> b - a = ratio (b - a) (c - a) *: (c - a) -> extension a b (ratio (b - a) (c - a)) = c.
proof: Proof. move=> neq_ab ratioP; apply /eqP; rewrite /extension eq_sym -subr_eq. case: (a =P c)=> [->|/eqP ?]; first by rewrite subrr ratiop0 invr0 scale0r. rewrite {2}ratioP scalerA -div1r -mulrC mulrA mulr1 divff ?scale1r ?eqxx //. by move/eqP: ratioP=> ?; rewrite ratio_neq0 // subr_eq add0r eq_sym. Qed.
imports: mathcomp.ssreflect.ssreflect mathcomp(ssrbool ssrfun ssrnat eqtype seq order) mathcomp(fintype bigop) mathcomp(ssralg ssrnum matrix) mathcomp(zmodp) GeoCoq.Axioms.tarski_axioms GeoCoq.Axioms.gupta_inspired_variant_axioms GeoCoq.Meta_theory.Models.gupta_inspired_to_tarski GeoCoq.Meta_theory.Counter_models.coplanarity GeoCoq.Meta_theory.Parallel_postulates.parallel_postulates GeoCoq.Tarski_dev.Ch12_parallel_inter_dec