proposition: Lemma ratio_bet a b c k1 k2 k3 : 0 < k1 -> 0 < k2 -> k1 < 1 -> 0 < k3 -> k3 < k1 + k2 - k1 * k2 -> b - a == ((k1 + k2 - k1 * k2) / k3)^-1 *: (c - a) -> bet a b c.
proof: Proof. move=> L1 L2 L3 L4 L5; rewrite /bet /betE. case: (a =P b)=> [->|/eqP ab_neq]. have: (0 < k1 + k2 - k1 * k2)=> [|L6]; first by rewrite (lt_trans L4). rewrite subrr eq_sym scalemx_eq0 subr_eq0 [c == b]eq_sym => /orP[HF|-> //]. rewrite lt0r in L4; rewrite lt0r in L6; move: HF L4 L6. by rewrite invr_eq0 mulf_eq0 invr_eq0 => /orP[/eqP->|/eqP->]; rewrite eqxx. by move=> ?; apply /orP; right; apply (ratio_betS ab_neq L1 L2 L3 L4). 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