-- Module: Mathlib.Geometry.Euclidean.MongePoint
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
t : Triangle ℝ P
i₁ i₂ : Fin 3
h : i₁ ≠ i₂
hu : Subset (α := Finset (Fin 3)) {i₁, i₂} univ
obtain :
  ∀ (i₃ : Fin 3) (hi₃ : Eq (α := Finset (Fin 3)) (univ \ {i₁, i₂}) {i₃}) (hi₃₁ : i₃ ≠ i₁) (hi₃₂ : i₃ ≠ i₂),
    -(∑ x : Fin (0 + 2 + 1),
              (∑ x_1 : Fin (0 + 2 + 1),
                  HSub.hSub (α := ℝ) (↑(HAdd.hAdd (α := ℕ) 0 1) : ℝ)⁻¹ (if x = i₁ ∨ x = i₂ then 1 else 0) *
                      HSub.hSub (α := ℝ) (↑(HAdd.hAdd (α := ℕ) 0 1) : ℝ)⁻¹ (if x_1 = i₁ ∨ x_1 = i₂ then 1 else 0) *
                    (dist (pointsWithCircumcenter t (PointsWithCircumcenterIndex.pointIndex x))
                        (pointsWithCircumcenter t (PointsWithCircumcenterIndex.pointIndex x_1)) *
                      dist (pointsWithCircumcenter t (PointsWithCircumcenterIndex.pointIndex x))
                        (pointsWithCircumcenter t (PointsWithCircumcenterIndex.pointIndex x_1))) +
                HSub.hSub (α := ℝ) (↑(HAdd.hAdd (α := ℕ) 0 1) : ℝ)⁻¹ (if x = i₁ ∨ x = i₂ then 1 else 0) *
                    (HDiv.hDiv (α := ℝ) (-2) (↑(HAdd.hAdd (α := ℕ) 0 1) : ℝ) - -1) *
                  (dist (pointsWithCircumcenter t (PointsWithCircumcenterIndex.pointIndex x))
                      (pointsWithCircumcenter t PointsWithCircumcenterIndex.circumcenterIndex) *
                    dist (pointsWithCircumcenter t (PointsWithCircumcenterIndex.pointIndex x))
                      (pointsWithCircumcenter t PointsWithCircumcenterIndex.circumcenterIndex))) +
            (∑ x : Fin (0 + 2 + 1),
                (HDiv.hDiv (α := ℝ) (-2) (↑(HAdd.hAdd (α := ℕ) 0 1) : ℝ) - -1) *
                    HSub.hSub (α := ℝ) (↑(HAdd.hAdd (α := ℕ) 0 1) : ℝ)⁻¹ (if x = i₁ ∨ x = i₂ then 1 else 0) *
                  (dist (pointsWithCircumcenter t PointsWithCircumcenterIndex.circumcenterIndex)
                      (pointsWithCircumcenter t (PointsWithCircumcenterIndex.pointIndex x)) *
                    dist (pointsWithCircumcenter t PointsWithCircumcenterIndex.circumcenterIndex)
                      (pointsWithCircumcenter t (PointsWithCircumcenterIndex.pointIndex x))) +
              (HDiv.hDiv (α := ℝ) (-2) (↑(HAdd.hAdd (α := ℕ) 0 1) : ℝ) - -1) *
                  (HDiv.hDiv (α := ℝ) (-2) (↑(HAdd.hAdd (α := ℕ) 0 1) : ℝ) - -1) *
                (dist (pointsWithCircumcenter t PointsWithCircumcenterIndex.circumcenterIndex)
                    (pointsWithCircumcenter t PointsWithCircumcenterIndex.circumcenterIndex) *
                  dist (pointsWithCircumcenter t PointsWithCircumcenterIndex.circumcenterIndex)
                    (pointsWithCircumcenter t PointsWithCircumcenterIndex.circumcenterIndex)))) /
        2 =
      circumradius t * circumradius t
⊢ -(∑ x : Fin (0 + 2 + 1),
            (∑ x_1 : Fin (0 + 2 + 1),
                HSub.hSub (α := ℝ) (↑(HAdd.hAdd (α := ℕ) 0 1) : ℝ)⁻¹ (if x = i₁ ∨ x = i₂ then 1 else 0) *
                    HSub.hSub (α := ℝ) (↑(HAdd.hAdd (α := ℕ) 0 1) : ℝ)⁻¹ (if x_1 = i₁ ∨ x_1 = i₂ then 1 else 0) *
                  (dist (pointsWithCircumcenter t (PointsWithCircumcenterIndex.pointIndex x))
                      (pointsWithCircumcenter t (PointsWithCircumcenterIndex.pointIndex x_1)) *
                    dist (pointsWithCircumcenter t (PointsWithCircumcenterIndex.pointIndex x))
                      (pointsWithCircumcenter t (PointsWithCircumcenterIndex.pointIndex x_1))) +
              HSub.hSub (α := ℝ) (↑(HAdd.hAdd (α := ℕ) 0 1) : ℝ)⁻¹ (if x = i₁ ∨ x = i₂ then 1 else 0) *
                  (HDiv.hDiv (α := ℝ) (-2) (↑(HAdd.hAdd (α := ℕ) 0 1) : ℝ) - -1) *
                (dist (pointsWithCircumcenter t (PointsWithCircumcenterIndex.pointIndex x))
                    (pointsWithCircumcenter t PointsWithCircumcenterIndex.circumcenterIndex) *
                  dist (pointsWithCircumcenter t (PointsWithCircumcenterIndex.pointIndex x))
                    (pointsWithCircumcenter t PointsWithCircumcenterIndex.circumcenterIndex))) +
          (∑ x : Fin (0 + 2 + 1),
              (HDiv.hDiv (α := ℝ) (-2) (↑(HAdd.hAdd (α := ℕ) 0 1) : ℝ) - -1) *
                  HSub.hSub (α := ℝ) (↑(HAdd.hAdd (α := ℕ) 0 1) : ℝ)⁻¹ (if x = i₁ ∨ x = i₂ then 1 else 0) *
                (dist (pointsWithCircumcenter t PointsWithCircumcenterIndex.circumcenterIndex)
                    (pointsWithCircumcenter t (PointsWithCircumcenterIndex.pointIndex x)) *
                  dist (pointsWithCircumcenter t PointsWithCircumcenterIndex.circumcenterIndex)
                    (pointsWithCircumcenter t (PointsWithCircumcenterIndex.pointIndex x))) +
            (HDiv.hDiv (α := ℝ) (-2) (↑(HAdd.hAdd (α := ℕ) 0 1) : ℝ) - -1) *
                (HDiv.hDiv (α := ℝ) (-2) (↑(HAdd.hAdd (α := ℕ) 0 1) : ℝ) - -1) *
              (dist (pointsWithCircumcenter t PointsWithCircumcenterIndex.circumcenterIndex)
                  (pointsWithCircumcenter t PointsWithCircumcenterIndex.circumcenterIndex) *
                dist (pointsWithCircumcenter t PointsWithCircumcenterIndex.circumcenterIndex)
                  (pointsWithCircumcenter t PointsWithCircumcenterIndex.circumcenterIndex)))) /
      2 =
    circumradius t * circumradius t

Tactic:
obtain ⟨i₃, hi₃, hi₃₁, hi₃₂⟩ :
      ∃ i₃, univ \ ({i₁, i₂} : Finset (Fin 3)) = {i₃} ∧ i₃ ≠ i₁ ∧ i₃ ≠ i₂ := by
    decide +revert

Premises used: Finset.univ, Insert.insert, Singleton.singleton