English
In an orthocentric system, any three distinct points are affinely independent. More precisely, if t is a triangle from the system and its vertices p0, p1, p2 lie in the system, then p0, p1, p2 are affinely independent in ℝ.
Русский
В ортогональной системе любые три различных точкиAffine независимы. Точка p0, p1, p2 образуют аффинно независимый треугольник.
LaTeX
$$$\\text{AffineIndependent}_{\\mathbb{R}}(p_0,p_1,p_2)$$$
Lean4
/-- Any three points in an orthocentric system are affinely independent. -/
theorem affineIndependent {s : Set P} (ho : OrthocentricSystem s) {p : Fin 3 → P} (hps : Set.range p ⊆ s)
(hpi : Function.Injective p) : AffineIndependent ℝ p :=
by
rcases ho with ⟨t, hto, hst⟩
rw [hst] at hps
rcases exists_dist_eq_circumradius_of_subset_insert_orthocenter hto hps hpi with ⟨c, _, hc⟩
exact Cospherical.affineIndependent ⟨c, t.circumradius, hc⟩ Set.Subset.rfl hpi
