/-- Suppose all distances from `p₁` and `p₂` to the points of a
simplex are equal, and that `p₁` and `p₂` lie in the affine span of
`p` with the vertices of that simplex.  Then `p₁` and `p₂` are equal
or reflections of each other in the affine span of the vertices of the
simplex. -/
theorem eq_or_eq_reflection_of_dist_eq {n : ℕ} {s : Simplex ℝ P n} {p p₁ p₂ : P} {r : ℝ}
    (hp₁ : p₁ ∈ affineSpan ℝ (insert p (Set.range s.points)))
    (hp₂ : p₂ ∈ affineSpan ℝ (insert p (Set.range s.points))) (h₁ : ∀ i, dist (s.points i) p₁ = r)
    (h₂ : ∀ i, dist (s.points i) p₂ = r) :
    p₁ = p₂ ∨ p₁ = reflection (affineSpan ℝ (Set.range s.points)) p₂ := by
  set span_s := affineSpan ℝ (Set.range s.points)
  have h₁' := s.orthogonalProjection_eq_circumcenter_of_dist_eq h₁
  have h₂' := s.orthogonalProjection_eq_circumcenter_of_dist_eq h₂
  rw [← affineSpan_insert_affineSpan, mem_affineSpan_insert_iff (orthogonalProjection_mem p)]
    at hp₁ hp₂
  obtain ⟨r₁, p₁o, hp₁o, hp₁⟩ := hp₁
  obtain ⟨r₂, p₂o, hp₂o, hp₂⟩ := hp₂
  obtain rfl : ↑(s.orthogonalProjectionSpan p₁) = p₁o := by
    subst hp₁
    exact s.coe_orthogonalProjection_vadd_smul_vsub_orthogonalProjection hp₁o
  rw [h₁'] at hp₁
  obtain rfl : ↑(s.orthogonalProjectionSpan p₂) = p₂o := by
    subst hp₂
    exact s.coe_orthogonalProjection_vadd_smul_vsub_orthogonalProjection hp₂o
  rw [h₂'] at hp₂
  have h : s.points 0 ∈ span_s := mem_affineSpan ℝ (Set.mem_range_self _)
  have hd₁ :
    dist p₁ s.circumcenter * dist p₁ s.circumcenter = r * r - s.circumradius * s.circumradius :=
    s.dist_circumcenter_sq_eq_sq_sub_circumradius h₁ h₁' h
  have hd₂ :
    dist p₂ s.circumcenter * dist p₂ s.circumcenter = r * r - s.circumradius * s.circumradius :=
    s.dist_circumcenter_sq_eq_sq_sub_circumradius h₂ h₂' h
  rw [← hd₂, hp₁, hp₂, dist_eq_norm_vsub V _ s.circumcenter, dist_eq_norm_vsub V _ s.circumcenter,
    vadd_vsub, vadd_vsub, ← real_inner_self_eq_norm_mul_norm, ← real_inner_self_eq_norm_mul_norm,
    real_inner_smul_left, real_inner_smul_left, real_inner_smul_right, real_inner_smul_right, ←
    mul_assoc, ← mul_assoc] at hd₁
  by_cases hp : p = s.orthogonalProjectionSpan p
  · rw [Simplex.orthogonalProjectionSpan] at hp
    rw [hp₁, hp₂, ← hp]
    simp only [true_or_iff, eq_self_iff_true, smul_zero, vsub_self]
  · have hz : ⟪p -ᵥ orthogonalProjection span_s p, p -ᵥ orthogonalProjection span_s p⟫ ≠ 0 := by
      simpa only [Ne.def, vsub_eq_zero_iff_eq, inner_self_eq_zero] using hp
    rw [mul_left_inj' hz, mul_self_eq_mul_self_iff] at hd₁
    rw [hp₁, hp₂]
    

Proof state:
case neg
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 p₁ p₂ : P
r : ℝ
h₁ : ∀ (i : Fin (n + 1)), dist (Simplex.points s i) p₁ = r
h₂ : ∀ (i : Fin (n + 1)), dist (Simplex.points s i) p₂ = r
span_s : AffineSubspace ℝ P := affineSpan ℝ (Set.range s.points)
h₁' : ↑((Simplex.orthogonalProjectionSpan s) p₁) = Simplex.circumcenter s
h₂' : ↑((Simplex.orthogonalProjectionSpan s) p₂) = Simplex.circumcenter s
r₁ r₂ : ℝ
hp₁o : ↑((Simplex.orthogonalProjectionSpan s) p₁) ∈ affineSpan ℝ (Set.range s.points)
hp₁ : p₁ = r₁ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p)) +ᵥ Simplex.circumcenter s
hp₂o : ↑((Simplex.orthogonalProjectionSpan s) p₂) ∈ affineSpan ℝ (Set.range s.points)
hp₂ : p₂ = r₂ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p)) +ᵥ Simplex.circumcenter s
h : Simplex.points s 0 ∈ span_s
hd₁ : r₁ = r₂ ∨ r₁ = -r₂
hd₂ :
  dist p₂ (Simplex.circumcenter s) * dist p₂ (Simplex.circumcenter s) =
    r * r - Simplex.circumradius s * Simplex.circumradius s
hp : ¬p = ↑((Simplex.orthogonalProjectionSpan s) p)
hz : inner (p -ᵥ ↑((orthogonalProjection span_s) p)) (p -ᵥ ↑((orthogonalProjection span_s) p)) ≠ 0
⊢ r₁ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p)) +ᵥ Simplex.circumcenter s =
      r₂ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p)) +ᵥ Simplex.circumcenter s ∨
    r₁ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p)) +ᵥ Simplex.circumcenter s =
      (reflection (affineSpan ℝ (Set.range s.points)))
        (r₂ • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p)) +ᵥ Simplex.circumcenter s)

Next tactic:
cases' hd₁ with hd₁ hd₁