English
If ps is cospherical and contained in an affine subspace s, then all simplices built from ps have the same circumsphere.
Русский
Если ps кососферично и содержится в аффинном подпространстве s, то все симплексы из ps имеют одну и ту же circumsphere.
LaTeX
$$$\exists c \in \text{Sphere } P, \forall sx : \text{Simplex } \mathbb{R} P\, n,\; (\text{range } sx.points \subseteq ps) \Rightarrow sx.circumsphere = c$$$
Lean4
/-- Two n-simplices among cospherical points in an n-dimensional
subspace have the same circumsphere. -/
theorem circumsphere_eq_of_cospherical_subset {s : AffineSubspace ℝ P} {ps : Set P} (h : ps ⊆ s) [Nonempty s] {n : ℕ}
[FiniteDimensional ℝ s.direction] (hd : finrank ℝ s.direction = n) (hc : Cospherical ps) {sx₁ sx₂ : Simplex ℝ P n}
(hsx₁ : Set.range sx₁.points ⊆ ps) (hsx₂ : Set.range sx₂.points ⊆ ps) : sx₁.circumsphere = sx₂.circumsphere :=
by
rcases exists_circumsphere_eq_of_cospherical_subset h hd hc with ⟨r, hr⟩
rw [hr sx₁ hsx₁, hr sx₂ hsx₂]
