-- Mathlib/Geometry/Euclidean/MongePoint.lean
theorem inner_mongePoint_vsub_face_centroid_vsub {n : ℕ} (s : Simplex ℝ P (n + 2))
    {i₁ i₂ : Fin (n + 3)} :
    ⟪s.mongePoint -ᵥ ({i₁, i₂}ᶜ : Finset (Fin (n + 3))).centroid ℝ s.points,
        s.points i₁ -ᵥ s.points i₂⟫ =
      0

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 + 2)
i₁ i₂ : Fin (n + 3)
h : ¬i₁ = i₂
hs :
  ∑ i : PointsWithCircumcenterIndex (n + 2), (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i = 0
fs : Finset (Fin (n + 3)) := {i₁, i₂}
hfs : ∀ i ∉ fs, i ≠ i₁ ∧ i ≠ i₂
⊢ -(∑ x ∈ fs,
                (if True ∨ i₁ = i₂ then (↑(n + 1))⁻¹ else 0) * ((if x = i₁ then 1 else 0) - if x = i₂ then 1 else 0) *
                  (dist (s.points i₁) (s.points x) * dist (s.points i₁) (s.points x)) +
              (if True ∨ i₁ = i₂ then (↑(n + 1))⁻¹ else 0) * (0 - 0) *
                (dist (s.points i₁) s.circumcenter * dist (s.points i₁) s.circumcenter) +
            (∑ x ∈ fs,
                (if i₂ = i₁ ∨ True then (↑(n + 1))⁻¹ else 0) * ((if x = i₁ then 1 else 0) - if x = i₂ then 1 else 0) *
                  (dist (s.points i₂) (s.points x) * dist (s.points i₂) (s.points x)) +
              (if i₂ = i₁ ∨ True then (↑(n + 1))⁻¹ else 0) * (0 - 0) *
                (dist (s.points i₂) s.circumcenter * dist (s.points i₂) s.circumcenter)) +
          (-2 / ↑(n + 1) * ((if True then 1 else 0) - if i₁ = i₂ then 1 else 0) *
                (dist s.circumcenter (s.points i₁) * dist s.circumcenter (s.points i₁)) +
              -2 / ↑(n + 1) * ((if i₂ = i₁ then 1 else 0) - if True then 1 else 0) *
                (dist s.circumcenter (s.points i₂) * dist s.circumcenter (s.points i₂)) +
            -2 / ↑(n + 1) * (0 - 0) * (dist s.circumcenter s.circumcenter * dist s.circumcenter s.circumcenter))) /
      2 =
    0

Tactic:
simp_rw [fs, sum_insert (not_mem_singleton.2 h), sum_singleton]

Result:
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 + 2)
i₁ i₂ : Fin (n + 3)
h : ¬i₁ = i₂
hs :
  ∑ i : PointsWithCircumcenterIndex (n + 2), (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i = 0
fs : Finset (Fin (n + 3)) := {i₁, i₂}
hfs : ∀ i ∉ fs, i ≠ i₁ ∧ i ≠ i₂
⊢ -((if True ∨ i₁ = i₂ then (↑(n + 1))⁻¹ else 0) * ((if True then 1 else 0) - if i₁ = i₂ then 1 else 0) *
                  (dist (s.points i₁) (s.points i₁) * dist (s.points i₁) (s.points i₁)) +
                (if True ∨ i₁ = i₂ then (↑(n + 1))⁻¹ else 0) * ((if i₂ = i₁ then 1 else 0) - if True then 1 else 0) *
                  (dist (s.points i₁) (s.points i₂) * dist (s.points i₁) (s.points i₂)) +
              (if True ∨ i₁ = i₂ then (↑(n + 1))⁻¹ else 0) * (0 - 0) *
                (dist (s.points i₁) s.circumcenter * dist (s.points i₁) s.circumcenter) +
            ((if i₂ = i₁ ∨ True then (↑(n + 1))⁻¹ else 0) * ((if True then 1 else 0) - if i₁ = i₂ then 1 else 0) *
                  (dist (s.points i₂) (s.points i₁) * dist (s.points i₂) (s.points i₁)) +
                (if i₂ = i₁ ∨ True then (↑(n + 1))⁻¹ else 0) * ((if i₂ = i₁ then 1 else 0) - if True then 1 else 0) *
                  (dist (s.points i₂) (s.points i₂) * dist (s.points i₂) (s.points i₂)) +
              (if i₂ = i₁ ∨ True then (↑(n + 1))⁻¹ else 0) * (0 - 0) *
                (dist (s.points i₂) s.circumcenter * dist (s.points i₂) s.circumcenter)) +
          (-2 / ↑(n + 1) * ((if True then 1 else 0) - if i₁ = i₂ then 1 else 0) *
                (dist s.circumcenter (s.points i₁) * dist s.circumcenter (s.points i₁)) +
              -2 / ↑(n + 1) * ((if i₂ = i₁ then 1 else 0) - if True then 1 else 0) *
                (dist s.circumcenter (s.points i₂) * dist s.circumcenter (s.points i₂)) +
            -2 / ↑(n + 1) * (0 - 0) * (dist s.circumcenter s.circumcenter * dist s.circumcenter s.circumcenter))) /
      2 =
    0

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 + 2)
i₁ i₂ : Fin (n + 3)
h : ¬i₁ = i₂
hs :
  ∑ i : PointsWithCircumcenterIndex (n + 2), (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i = 0
⊢ -(∑ i : Fin (n + 2 + 1),
            ∑ i₂_1 : PointsWithCircumcenterIndex (n + 2),
              mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂ (pointIndex i) *
                  (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i₂_1 *
                (dist (s.pointsWithCircumcenter (pointIndex i)) (s.pointsWithCircumcenter i₂_1) *
                  dist (s.pointsWithCircumcenter (pointIndex i)) (s.pointsWithCircumcenter i₂_1)) +
          ∑ i₂_1 : PointsWithCircumcenterIndex (n + 2),
            mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂ circumcenterIndex *
                (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i₂_1 *
              (dist s.circumcenter (s.pointsWithCircumcenter i₂_1) *
                dist s.circumcenter (s.pointsWithCircumcenter i₂_1))) /
      2 =
    0

Tactic:
simp only [mongePointVSubFaceCentroidWeightsWithCircumcenter, pointsWithCircumcenter_point]

Result:
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 + 2)
i₁ i₂ : Fin (n + 3)
h : ¬i₁ = i₂
hs :
  ∑ i : PointsWithCircumcenterIndex (n + 2), (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i = 0
⊢ -(∑ x : Fin (n + 2 + 1),
            ∑ x_1 : PointsWithCircumcenterIndex (n + 2),
              (if x = i₁ ∨ x = i₂ then (↑(n + 1))⁻¹ else 0) *
                  (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) x_1 *
                (dist (s.points x) (s.pointsWithCircumcenter x_1) * dist (s.points x) (s.pointsWithCircumcenter x_1)) +
          ∑ x : PointsWithCircumcenterIndex (n + 2),
            -2 / ↑(n + 1) * (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) x *
              (dist s.circumcenter (s.pointsWithCircumcenter x) * dist s.circumcenter (s.pointsWithCircumcenter x))) /
      2 =
    0