Module: Mathlib.Geometry.Euclidean.MongePoint

Goal state before:
pp: 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
t : Triangle ℝ P
i₁ i₂ : Fin 3
h : i₁ ≠ i₂
hu : {i₁, i₂} ⊆ univ
⊢ -(∑ x,
            (∑ x_1,
                ((↑(0 + 1))⁻¹ - if x = i₁ ∨ x = i₂ then 1 else 0) *
                    ((↑(0 + 1))⁻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))) +
              ((↑(0 + 1))⁻¹ - if x = i₁ ∨ x = i₂ then 1 else 0) * (-2 / ↑(0 + 1) - -1) *
                (dist (pointsWithCircumcenter t (PointsWithCircumcenterIndex.pointIndex x))
                    (pointsWithCircumcenter t PointsWithCircumcenterIndex.circumcenterIndex) *
                  dist (pointsWithCircumcenter t (PointsWithCircumcenterIndex.pointIndex x))
                    (pointsWithCircumcenter t PointsWithCircumcenterIndex.circumcenterIndex))) +
          (∑ x,
              (-2 / ↑(0 + 1) - -1) * ((↑(0 + 1))⁻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))) +
            (-2 / ↑(0 + 1) - -1) * (-2 / ↑(0 + 1) - -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

Goal 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
t : Triangle ℝ P
i₁ i₂ : Fin 3
h : i₁ ≠ i₂
hu : {i₁, i₂} ⊆ univ
i₃ : Fin 3
hi₃ : univ \ {i₁, i₂} = {i₃}
hi₃₁ : i₃ ≠ i₁
hi₃₂ : i₃ ≠ i₂
⊢ -(∑ x,
            (∑ x_1,
                ((↑(0 + 1))⁻¹ - if x = i₁ ∨ x = i₂ then 1 else 0) *
                    ((↑(0 + 1))⁻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))) +
              ((↑(0 + 1))⁻¹ - if x = i₁ ∨ x = i₂ then 1 else 0) * (-2 / ↑(0 + 1) - -1) *
                (dist (pointsWithCircumcenter t (PointsWithCircumcenterIndex.pointIndex x))
                    (pointsWithCircumcenter t PointsWithCircumcenterIndex.circumcenterIndex) *
                  dist (pointsWithCircumcenter t (PointsWithCircumcenterIndex.pointIndex x))
                    (pointsWithCircumcenter t PointsWithCircumcenterIndex.circumcenterIndex))) +
          (∑ x,
              (-2 / ↑(0 + 1) - -1) * ((↑(0 + 1))⁻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))) +
            (-2 / ↑(0 + 1) - -1) * (-2 / ↑(0 + 1) - -1) *
              (dist (pointsWithCircumcenter t PointsWithCircumcenterIndex.circumcenterIndex)
                  (pointsWithCircumcenter t PointsWithCircumcenterIndex.circumcenterIndex) *
                dist (pointsWithCircumcenter t PointsWithCircumcenterIndex.circumcenterIndex)
                  (pointsWithCircumcenter t PointsWithCircumcenterIndex.circumcenterIndex)))) /
      2 =
    circumradius t * circumradius t