Theorem: Affine.Simplex.eq_circumcenter_of_dist_eq
File: Mathlib/Geometry/Euclidean/Circumcenter.lean

Proof steps:
state_before: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P n
p : P
hp : p ∈ affineSpan ℝ (Set.range s.points)
r : ℝ
hr : ∀ (i : Fin (n + 1)), dist (s.points i) p = r
⊢ p = s.circumcenter
state_after: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P n
p : P
hp : p ∈ affineSpan ℝ (Set.range s.points)
r : ℝ
hr : ∀ (i : Fin (n + 1)), dist (s.points i) p = r
h :
  { center := p, radius := r }.center ∈ affineSpan ℝ (Set.range s.points) ∧
      Set.range s.points ⊆ Metric.sphere { center := p, radius := r }.center { center := p, radius := r }.radius →
    { center := p, radius := r } = s.circumsphere
⊢ p = s.circumcenter
tactic: have h := s.circumsphere_unique_dist_eq.2 ⟨p, r⟩
state_before: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P n
p : P
hp : p ∈ affineSpan ℝ (Set.range s.points)
r : ℝ
hr : ∀ (i : Fin (n + 1)), dist (s.points i) p = r
h :
  { center := p, radius := r }.center ∈ affineSpan ℝ (Set.range s.points) ∧
      Set.range s.points ⊆ Metric.sphere { center := p, radius := r }.center { center := p, radius := r }.radius →
    { center := p, radius := r } = s.circumsphere
⊢ p = s.circumcenter
state_after: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P n
p : P
hp : p ∈ affineSpan ℝ (Set.range s.points)
r : ℝ
hr : ∀ (i : Fin (n + 1)), dist (s.points i) p = r
h : Set.range s.points ⊆ Metric.sphere p r → p = s.circumsphere.center ∧ r = s.circumsphere.radius
⊢ p = s.circumcenter
tactic: simp only [hp, hr, forall_const, eq_self_iff_true, subset_sphere, Sphere.ext_iff,
    Set.forall_mem_range, mem_sphere, true_and] at h
state_before: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P n
p : P
hp : p ∈ affineSpan ℝ (Set.range s.points)
r : ℝ
hr : ∀ (i : Fin (n + 1)), dist (s.points i) p = r
h : Set.range s.points ⊆ Metric.sphere p r → p = s.circumsphere.center ∧ r = s.circumsphere.radius
⊢ p = s.circumcenter
state_after: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P n
p : P
hp : p ∈ affineSpan ℝ (Set.range s.points)
r : ℝ
hr : ∀ (i : Fin (n + 1)), dist (s.points i) p = r
h :
  (∀ p_1 ∈ Set.range s.points, p_1 ∈ { center := p, radius := r }) →
    p = s.circumsphere.center ∧ r = s.circumsphere.radius
⊢ p = s.circumcenter
tactic: rw [subset_sphere (s := ⟨p, r⟩)] at h
state_before: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P n
p : P
hp : p ∈ affineSpan ℝ (Set.range s.points)
r : ℝ
hr : ∀ (i : Fin (n + 1)), dist (s.points i) p = r
h :
  (∀ p_1 ∈ Set.range s.points, p_1 ∈ { center := p, radius := r }) →
    p = s.circumsphere.center ∧ r = s.circumsphere.radius
⊢ p = s.circumcenter
state_after: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P n
p : P
hp : p ∈ affineSpan ℝ (Set.range s.points)
r : ℝ
hr : ∀ (i : Fin (n + 1)), dist (s.points i) p = r
h : p = s.circumsphere.center ∧ r = s.circumsphere.radius
⊢ p = s.circumcenter
tactic: simp only [hp, hr, forall_const, eq_self_iff_true, subset_sphere, Sphere.ext_iff,
    Set.forall_mem_range, mem_sphere, true_and] at h
state_before: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P n
p : P
hp : p ∈ affineSpan ℝ (Set.range s.points)
r : ℝ
hr : ∀ (i : Fin (n + 1)), dist (s.points i) p = r
h : p = s.circumsphere.center ∧ r = s.circumsphere.radius
⊢ p = s.circumcenter
state_after: no goals
tactic: exact h.1