English
For a 0-simplex, the circumcenter equals its unique vertex.
Русский
Для 0-симплексa circumcenter совпадает с единственной вершиной.
LaTeX
$$$(s : Simplex ℝ P 0).circumcenter = (s.points 0)$$$
Lean4
/-- The circumcenter of a 0-simplex equals its unique point. -/
theorem circumcenter_eq_point (s : Simplex ℝ P 0) (i : Fin 1) : s.circumcenter = s.points i :=
by
have h := s.circumcenter_mem_affineSpan
have : Unique (Fin 1) := ⟨⟨0, by decide⟩, fun a => by simp only [Fin.eq_zero]⟩
simp only [Set.range_unique, AffineSubspace.mem_affineSpan_singleton] at h
rw [h]
congr
simp only [eq_iff_true_of_subsingleton]