mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Euclidean.NinePointCircle

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
n : ℕ
s : Simplex ℝ P n
i : Fin (n + 1)
⊢ s.points i -ᵥ s.mongePoint - (↑n)⁻¹ • (s.points i -ᵥ s.mongePoint) = ((↑n - 1) / ↑n) • (s.points i -ᵥ s.mongePoint)

Tactic:
]

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
n : ℕ
s : Simplex ℝ P n
i : Fin (n + 1)
⊢ s.points i -ᵥ s.mongePoint - (↑n)⁻¹ • (s.points i -ᵥ s.mongePoint) = ((↑n - 1) / ↑n) • (s.points i -ᵥ s.mongePoint)
Module: Mathlib.Topology.MetricSpace.Similarity

Goal state before:
pp: ι : Type u_1
P₁ : Type u_3
P₂ : Type u_4
P₃ : Type u_5
v₁ : ι → P₁
v₂ : ι → P₂
inst✝⁴ : PseudoEMetricSpace P₁
inst✝³ : PseudoEMetricSpace P₂
inst✝² : PseudoEMetricSpace P₃
F : Type u_6
inst✝¹ : FunLike F P₂ P₃
inst✝ : DilationClass F P₂ P₃
f : F
⊢ v₁ ∼ v₂ ↔ v₁ ∼ v₂

Tactic:
]

Goal state after:
Module: Mathlib.Geometry.Euclidean.NinePointCircle

Goal state before:
pp: case pos
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
s : Simplex ℝ P 0
i : Fin (0 + 1)
hrange : Set.range s.points = {s.points i}
hmonge : s.mongePoint = s.points i
⊢ s.points i -ᵥ s.mongePoint - (↑0)⁻¹ • (s.points i -ᵥ s.mongePoint) = ((↑0 - 1) / ↑0) • (s.points i -ᵥ s.mongePoint)

Tactic:
]

Goal state after:
case pos
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
s : Simplex ℝ P 0
i : Fin (0 + 1)
hrange : Set.range s.points = {s.points i}
hmonge : s.mongePoint = s.points i
⊢ s.points i -ᵥ s.mongePoint - (↑0)⁻¹ • (s.points i -ᵥ s.mongePoint) = ((↑0 - 1) / ↑0) • (s.points i -ᵥ s.mongePoint)
Module: Mathlib.Geometry.Euclidean.NinePointCircle

Goal state before:
pp: 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
i : Fin (n + 1)
hn : ¬n = 0
⊢ s.points i -ᵥ s.mongePoint - (↑n)⁻¹ • (s.points i -ᵥ s.mongePoint) =
    s.points i -ᵥ s.mongePoint - (↑n)⁻¹ • (s.points i -ᵥ s.mongePoint)

Tactic:
]

Goal state after:
Module: Mathlib.Geometry.Euclidean.NinePointCircle

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
n : ℕ
hn : NeZero n
s : Simplex ℝ P n
i : Fin (n + 1)
⊢ ⅟2 • (s.faceOppositeCentroid i -ᵥ s.circumcenter) +
      ⅟2 •
        (((↑n)⁻¹ • (s.points i -ᵥ ((↑(n + 1) / ↑(n - 1)) • (s.centroid -ᵥ s.circumcenter) +ᵥ s.circumcenter)) +ᵥ
            (↑(n + 1) / ↑(n - 1)) • (s.centroid -ᵥ s.circumcenter) +ᵥ s.circumcenter) -ᵥ
          s.circumcenter) =
    ((↑n + 1) / ↑n) • (s.centroid -ᵥ s.circumcenter)

Tactic:
]

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
n : ℕ
hn : NeZero n
s : Simplex ℝ P n
i : Fin (n + 1)
⊢ ⅟2 • (s.faceOppositeCentroid i -ᵥ s.circumcenter) +
      ⅟2 •
        (((↑n)⁻¹ • (s.points i -ᵥ ((↑(n + 1) / ↑(n - 1)) • (s.centroid -ᵥ s.circumcenter) +ᵥ s.circumcenter)) +ᵥ
            (↑(n + 1) / ↑(n - 1)) • (s.centroid -ᵥ s.circumcenter) +ᵥ s.circumcenter) -ᵥ
          s.circumcenter) =
    ((↑n + 1) / ↑n) • (s.centroid -ᵥ s.circumcenter)
Module: Mathlib.Topology.MetricSpace.Similarity

Goal state before:
pp: ι : Type u_1
P₁ : Type u_3
P₂ : Type u_4
v₁ : ι → P₁
v₂ : ι → P₂
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
x✝² : ℝ≥0
x✝¹ x✝ : ι
⊢ ↑(nndist (v₁ x✝¹) (v₁ x✝)) = ↑x✝² * ↑(nndist (v₂ x✝¹) (v₂ x✝)) ↔
    nndist (v₁ x✝¹) (v₁ x✝) = x✝² * nndist (v₂ x✝¹) (v₂ x✝)

Tactic:
]

Goal state after:
ι : Type u_1
P₁ : Type u_3
P₂ : Type u_4
v₁ : ι → P₁
v₂ : ι → P₂
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
x✝² : ℝ≥0
x✝¹ x✝ : ι
⊢ ↑(nndist (v₁ x✝¹) (v₁ x✝)) = ↑x✝² * ↑(nndist (v₂ x✝¹) (v₂ x✝)) ↔
    nndist (v₁ x✝¹) (v₁ x✝) = x✝² * nndist (v₂ x✝¹) (v₂ x✝)
Module: Mathlib.Topology.MetricSpace.Similarity

Goal state before:
pp: ι : Type u_1
P₁ : Type u_3
P₂ : Type u_4
v₁ : ι → P₁
v₂ : ι → P₂
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
x✝² : ℝ≥0
x✝¹ x✝ : ι
⊢ nndist (v₁ x✝¹) (v₁ x✝) = x✝² * nndist (v₂ x✝¹) (v₂ x✝) ↔
    ↑(nndist (v₁ x✝¹) (v₁ x✝)) = ↑x✝² * ↑(nndist (v₂ x✝¹) (v₂ x✝))

Tactic:
]

Goal state after:
ι : Type u_1
P₁ : Type u_3
P₂ : Type u_4
v₁ : ι → P₁
v₂ : ι → P₂
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
x✝² : ℝ≥0
x✝¹ x✝ : ι
⊢ nndist (v₁ x✝¹) (v₁ x✝) = x✝² * nndist (v₂ x✝¹) (v₂ x✝) ↔
    ↑(nndist (v₁ x✝¹) (v₁ x✝)) = ↑x✝² * ↑(nndist (v₂ x✝¹) (v₂ x✝))
Module: Mathlib.Topology.MetricSpace.Similarity

Goal state before:
pp: ι : Type u_1
P₁ : Type u_3
P₂ : Type u_4
v₁ : ι → P₁
v₂ : ι → P₂
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
⊢ (∃ r, r ≠ 0 ∧ ∀ (i₁ i₂ : ι), dist (v₁ i₁) (v₁ i₂) = ↑r * dist (v₂ i₁) (v₂ i₂)) ↔
    ∃ r, 0 < r ∧ ∀ (i₁ i₂ : ι), dist (v₁ i₁) (v₁ i₂) = r * dist (v₂ i₁) (v₂ i₂)

Tactic:
]

Goal state after:
ι : Type u_1
P₁ : Type u_3
P₂ : Type u_4
v₁ : ι → P₁
v₂ : ι → P₂
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
⊢ (∃ r, r ≠ 0 ∧ ∀ (i₁ i₂ : ι), dist (v₁ i₁) (v₁ i₂) = ↑r * dist (v₂ i₁) (v₂ i₂)) ↔
    ∃ r, 0 < r ∧ ∀ (i₁ i₂ : ι), dist (v₁ i₁) (v₁ i₂) = r * dist (v₂ i₁) (v₂ i₂)
Module: Mathlib.Geometry.Euclidean.NinePointCircle

Goal state before:
pp: case pos
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
hn : NeZero 1
s : Simplex ℝ P 1
i : Fin (1 + 1)
⊢ 2⁻¹ • ((↑1)⁻¹ • (s.centroid -ᵥ s.points i) + (s.points i -ᵥ s.centroid)) = 0

Tactic:
]

Goal state after:
case pos
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
hn : NeZero 1
s : Simplex ℝ P 1
i : Fin (1 + 1)
⊢ 2⁻¹ • ((↑1)⁻¹ • (s.centroid -ᵥ s.points i) + (s.points i -ᵥ s.centroid)) = 0
Module: Mathlib.Geometry.Euclidean.NinePointCircle

Goal state before:
pp: 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
hn : NeZero n
s : Simplex ℝ P n
i : Fin (n + 1)
hn1 : ¬n = 1
hltn : 1 < n
hnsub1 : ↑(n - 1) = ↑n - 1
⊢ ⅟2 •
      ((↑n)⁻¹ • (s.centroid -ᵥ s.circumcenter) + (s.centroid -ᵥ s.circumcenter) +
        (↑(n + 1) / ↑n) • (s.centroid -ᵥ s.circumcenter)) =
    ((↑n + 1) / ↑n) • (s.centroid -ᵥ s.circumcenter)

Tactic:
]

Goal state after:
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
hn : NeZero n
s : Simplex ℝ P n
i : Fin (n + 1)
hn1 : ¬n = 1
hltn : 1 < n
hnsub1 : ↑(n - 1) = ↑n - 1
⊢ ⅟2 •
      ((↑n)⁻¹ • (s.centroid -ᵥ s.circumcenter) + (s.centroid -ᵥ s.circumcenter) +
        (↑(n + 1) / ↑n) • (s.centroid -ᵥ s.circumcenter)) =
    ((↑n + 1) / ↑n) • (s.centroid -ᵥ s.circumcenter)
Module: Mathlib.Geometry.Euclidean.MongePoint

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MetricSpace P
inst✝⁴ : NormedAddTorsor V P
V₂ : Type u_3
P₂ : Type u_4
inst✝³ : NormedAddCommGroup V₂
inst✝² : InnerProductSpace ℝ V₂
inst✝¹ : MetricSpace P₂
inst✝ : NormedAddTorsor V₂ P₂
n : ℕ
s : Simplex ℝ P n
f : P →ᵃⁱ[ℝ] P₂
⊢ (↑(n + 1) / ↑(n - 1)) • ((s.map f.toAffineMap ⋯).centroid -ᵥ (s.map f.toAffineMap ⋯).circumcenter) +ᵥ
      (s.map f.toAffineMap ⋯).circumcenter =
    f ((↑(n + 1) / ↑(n - 1)) • (s.centroid -ᵥ s.circumcenter) +ᵥ s.circumcenter)

Tactic:
]

Goal state after:
V : Type u_1
P : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MetricSpace P
inst✝⁴ : NormedAddTorsor V P
V₂ : Type u_3
P₂ : Type u_4
inst✝³ : NormedAddCommGroup V₂
inst✝² : InnerProductSpace ℝ V₂
inst✝¹ : MetricSpace P₂
inst✝ : NormedAddTorsor V₂ P₂
n : ℕ
s : Simplex ℝ P n
f : P →ᵃⁱ[ℝ] P₂
⊢ (↑(n + 1) / ↑(n - 1)) • ((s.map f.toAffineMap ⋯).centroid -ᵥ (s.map f.toAffineMap ⋯).circumcenter) +ᵥ
      (s.map f.toAffineMap ⋯).circumcenter =
    f ((↑(n + 1) / ↑(n - 1)) • (s.centroid -ᵥ s.circumcenter) +ᵥ s.circumcenter)
Module: Mathlib.Geometry.Euclidean.NinePointCircle

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
n : ℕ
hn : NeZero n
s : Simplex ℝ P n
i : Fin (n + 1)
hn1 : ¬n = 1
hltn : 1 < n
hnsub1 : ↑(n - 1) = ↑n - 1
⊢ (↑n)⁻¹ • (s.centroid -ᵥ s.circumcenter) + (s.centroid -ᵥ s.circumcenter) =
    (↑n)⁻¹ • (s.centroid -ᵥ s.circumcenter) + (s.centroid -ᵥ s.circumcenter)

Tactic:
]

Goal state after:
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
n : ℕ
s : Simplex ℝ P (n + 2)
⊢ ((n + 1) • (↑(n + 2 + 1) / ↑(n + 2 - 1))) • (Finset.centroid ℝ univ s.points -ᵥ s.circumcenter) =
    ∑ i, (s.points i -ᵥ s.circumcenter)

Tactic:
]

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
n : ℕ
s : Simplex ℝ P (n + 2)
⊢ ((n + 1) • (↑(n + 2 + 1) / ↑(n + 2 - 1))) • (Finset.centroid ℝ univ s.points -ᵥ s.circumcenter) =
    ∑ i, (s.points i -ᵥ s.circumcenter)
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
n : ℕ
s : Simplex ℝ P (n + 2)
h : Invertible (↑n + 2 + 1)
⊢ Finset.centroid ℝ univ s.points -ᵥ s.circumcenter = ∑ x, ⅟(↑n + 2 + 1) • (s.points x -ᵥ s.circumcenter)

Tactic:
]

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
n : ℕ
s : Simplex ℝ P (n + 2)
h : Invertible (↑n + 2 + 1)
⊢ Finset.centroid ℝ univ s.points -ᵥ s.circumcenter = ∑ x, ⅟(↑n + 2 + 1) • (s.points x -ᵥ s.circumcenter)
Module: Mathlib.Geometry.Euclidean.NinePointCircle

Goal state before:
pp: 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
hn : NeZero n
s : Simplex ℝ P n
i : Fin (n + 1)
hn1 : ¬n = 1
hltn : 1 < n
hnsub1 : ↑(n - 1) = ↑n - 1
this :
  (↑n)⁻¹ • (s.centroid -ᵥ s.circumcenter) + (s.centroid -ᵥ s.circumcenter) =
    ((↑n + 1) / ↑n) • (s.centroid -ᵥ s.circumcenter)
⊢ (⅟2 * 2) • ((↑n + 1) / ↑n) • (s.centroid -ᵥ s.circumcenter) = ((↑n + 1) / ↑n) • (s.centroid -ᵥ s.circumcenter)

Tactic:
]

Goal state after:
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
hn : NeZero n
s : Simplex ℝ P n
i : Fin (n + 1)
hn1 : ¬n = 1
hltn : 1 < n
hnsub1 : ↑(n - 1) = ↑n - 1
this :
  (↑n)⁻¹ • (s.centroid -ᵥ s.circumcenter) + (s.centroid -ᵥ s.circumcenter) =
    ((↑n + 1) / ↑n) • (s.centroid -ᵥ s.circumcenter)
⊢ (⅟2 * 2) • ((↑n + 1) / ↑n) • (s.centroid -ᵥ s.circumcenter) = ((↑n + 1) / ↑n) • (s.centroid -ᵥ s.circumcenter)
Module: Mathlib.Topology.MetricSpace.Similarity

Goal state before:
pp: case h.«0».«0»
P₁ : Type u_3
P₂ : Type u_4
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
a b c : P₁
a' b' c' : P₂
h_ne : dist a b ≠ 0
h_ne' : dist a' b' ≠ 0
heq1 : dist a b * dist b' c' = dist b c * dist a' b'
heq2 : dist a b * dist c' a' = dist c a * dist a' b'
r : ℝ := dist a b / dist a' b'
hr : r = dist a b / dist a' b'
hr_pos : 0 < r
hij : (fun i ↦ i) ⟨0, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩
⊢ 0 = 0

Tactic:
]

Goal state after:
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
n : ℕ
s : Simplex ℝ P (n + 2)
h : Invertible (↑n + 2 + 1)
⊢ (affineCombination ℝ univ s.points) (centroidWeights ℝ univ) -ᵥ s.circumcenter =
    ∑ x, ⅟(↑n + 2 + 1) • (s.points x -ᵥ s.circumcenter)

Tactic:
]

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
n : ℕ
s : Simplex ℝ P (n + 2)
h : Invertible (↑n + 2 + 1)
⊢ (affineCombination ℝ univ s.points) (centroidWeights ℝ univ) -ᵥ s.circumcenter =
    ∑ x, ⅟(↑n + 2 + 1) • (s.points x -ᵥ s.circumcenter)
Module: Mathlib.Geometry.Euclidean.NinePointCircle

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
n : ℕ
inst✝ : NeZero n
s : Simplex ℝ P n
i : Fin (n + 1)
⊢ ∠ (s.faceOppositeCentroid i)
      (↑((orthogonalProjection (affineSpan ℝ (Set.range (s.faceOpposite i).points))) (s.eulerPoint i)))
      (s.eulerPoint i) =
    Real.pi / 2

Tactic:
]

Goal state after:
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
n : ℕ
inst✝ : NeZero n
s : Simplex ℝ P n
i : Fin (n + 1)
⊢ ∠ (s.faceOppositeCentroid i)
      (↑((orthogonalProjection (affineSpan ℝ (Set.range (s.faceOpposite i).points))) (s.eulerPoint i)))
      (s.eulerPoint i) =
    Real.pi / 2
Module: Mathlib.Geometry.Euclidean.NinePointCircle

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
s : Triangle ℝ P
i : Fin 3
⊢ ((↑2 - 1) / ↑2) • (s.points i -ᵥ Simplex.mongePoint s) =
    ⅟2 • (s.points i -ᵥ Simplex.mongePoint s) + ⅟2 • (s.points i -ᵥ s.points i)

Tactic:
]

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
s : Triangle ℝ P
i : Fin 3
⊢ ((↑2 - 1) / ↑2) • (s.points i -ᵥ Simplex.mongePoint s) =
    ⅟2 • (s.points i -ᵥ Simplex.mongePoint s) + ⅟2 • (s.points i -ᵥ s.points i)
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
n : ℕ
s : Simplex ℝ P (n + 2)
h : Invertible (↑n + 2 + 1)
⊢ ∑ i, centroidWeights ℝ univ i • (s.points i -ᵥ s.circumcenter) = ∑ x, ⅟(↑n + 2 + 1) • (s.points x -ᵥ s.circumcenter)

Tactic:
]

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
n : ℕ
s : Simplex ℝ P (n + 2)
h : Invertible (↑n + 2 + 1)
⊢ ∑ i, centroidWeights ℝ univ i • (s.points i -ᵥ s.circumcenter) = ∑ x, ⅟(↑n + 2 + 1) • (s.points x -ᵥ s.circumcenter)
Module: Mathlib.Geometry.Euclidean.NinePointCircle

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
s : Triangle ℝ P
i : Fin 3
⊢ ↑((Simplex.faceOpposite s i).orthogonalProjectionSpan (s.points i)) =
    ↑((Simplex.faceOpposite s i).orthogonalProjectionSpan (Simplex.eulerPoint s i))

Tactic:
]

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
s : Triangle ℝ P
i : Fin 3
⊢ ↑((Simplex.faceOpposite s i).orthogonalProjectionSpan (s.points i)) =
    ↑((Simplex.faceOpposite s i).orthogonalProjectionSpan (Simplex.eulerPoint s i))
Module: Mathlib.Topology.MetricSpace.Similarity

Goal state before:
pp: case h.«1».«1»
P₁ : Type u_3
P₂ : Type u_4
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
a b c : P₁
a' b' c' : P₂
h_ne : dist a b ≠ 0
h_ne' : dist a' b' ≠ 0
heq1 : dist a b * dist b' c' = dist b c * dist a' b'
heq2 : dist a b * dist c' a' = dist c a * dist a' b'
r : ℝ := dist a b / dist a' b'
hr : r = dist a b / dist a' b'
hr_pos : 0 < r
hij : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Geometry.Euclidean.NinePointCircle

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
s : Triangle ℝ P
i : Fin 3
⊢ s.points i -ᵥ s.orthocenter ∈ (vectorSpan ℝ (s.points '' {i}ᶜ))ᗮ

Tactic:
]

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
s : Triangle ℝ P
i : Fin 3
⊢ s.points i -ᵥ s.orthocenter ∈ (vectorSpan ℝ (s.points '' {i}ᶜ))ᗮ
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
n : ℕ
s : Simplex ℝ P n
S : AffineSubspace ℝ P
hS : affineSpan ℝ (Set.range s.points) ≤ S
this : Nonempty ↥S
⊢ ↑((↑(n + 1) / ↑(n - 1)) • ((s.restrict S hS).centroid -ᵥ (s.restrict S hS).circumcenter) +ᵥ
        (s.restrict S hS).circumcenter) =
    (↑(n + 1) / ↑(n - 1)) • (s.centroid -ᵥ s.circumcenter) +ᵥ s.circumcenter

Tactic:
]

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
n : ℕ
s : Simplex ℝ P n
S : AffineSubspace ℝ P
hS : affineSpan ℝ (Set.range s.points) ≤ S
this : Nonempty ↥S
⊢ ↑((↑(n + 1) / ↑(n - 1)) • ((s.restrict S hS).centroid -ᵥ (s.restrict S hS).circumcenter) +ᵥ
        (s.restrict S hS).circumcenter) =
    (↑(n + 1) / ↑(n - 1)) • (s.centroid -ᵥ s.circumcenter) +ᵥ s.circumcenter
Module: Mathlib.Geometry.Euclidean.Simplex

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
n : ℕ
s : Simplex ℝ P n
i₁ i₂ i₃ : Fin (n + 1)
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
r : ℝ
hr : ∀ (i j : Fin (n + 1)), i ≠ j → dist (s.points i) (s.points j) = r
⊢ Real.arccos
      ((‖s.points i₁ -ᵥ s.points i₂‖ * ‖s.points i₁ -ᵥ s.points i₂‖ +
              ‖s.points i₃ -ᵥ s.points i₂‖ * ‖s.points i₃ -ᵥ s.points i₂‖ -
            ‖s.points i₁ -ᵥ s.points i₂ - (s.points i₃ -ᵥ s.points i₂)‖ *
              ‖s.points i₁ -ᵥ s.points i₂ - (s.points i₃ -ᵥ s.points i₂)‖) /
          2 /
        (‖s.points i₁ -ᵥ s.points i₂‖ * ‖s.points i₃ -ᵥ s.points i₂‖)) =
    π / 3

Tactic:
]

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
n : ℕ
s : Simplex ℝ P n
i₁ i₂ i₃ : Fin (n + 1)
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
r : ℝ
hr : ∀ (i j : Fin (n + 1)), i ≠ j → dist (s.points i) (s.points j) = r
⊢ Real.arccos
      ((‖s.points i₁ -ᵥ s.points i₂‖ * ‖s.points i₁ -ᵥ s.points i₂‖ +
              ‖s.points i₃ -ᵥ s.points i₂‖ * ‖s.points i₃ -ᵥ s.points i₂‖ -
            ‖s.points i₁ -ᵥ s.points i₂ - (s.points i₃ -ᵥ s.points i₂)‖ *
              ‖s.points i₁ -ᵥ s.points i₂ - (s.points i₃ -ᵥ s.points i₂)‖) /
          2 /
        (‖s.points i₁ -ᵥ s.points i₂‖ * ‖s.points i₃ -ᵥ s.points i₂‖)) =
    π / 3
Module: Mathlib.Geometry.Euclidean.Similarity

Goal state before:
pp: V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst✝⁷ : NormedAddCommGroup V₁
inst✝⁶ : NormedAddCommGroup V₂
inst✝⁵ : InnerProductSpace ℝ V₁
inst✝⁴ : InnerProductSpace ℝ V₂
inst✝³ : MetricSpace P₁
inst✝² : MetricSpace P₂
inst✝¹ : NormedAddTorsor V₁ P₁
inst✝ : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h_not_col : ¬Collinear ℝ {a, b, c}
h₁ : ∠ a b c = ∠ a' b' c'
h₂ : ∠ b c a = ∠ b' c' a'
hne_pi_div_two : ∠ a b c ≠ Real.pi / 2 ∨ ∠ b c a ≠ Real.pi / 2
not_all_eq : a' ≠ b' ∨ b' ≠ c' ∨ a' ≠ c'
h_not_col' : ¬Collinear ℝ {a', b', c'}
h_pos1 : 0 < dist a b
h_pos1' : 0 < dist a' b'
h_pos2 : 0 < dist b c
h_pos2' : 0 < dist b' c'
h₃ : ∠ c a b = ∠ c' a' b'
h_sin_ne1 : Real.sin (∠ b c a) ≠ 0
h_sin_ne2 : Real.sin (∠ c a b) ≠ 0
h_sin1 : Real.sin (∠ c a b) = Real.sin (∠ b c a) * dist b c / dist a b
h_sin1' : Real.sin (∠ c' a' b') = Real.sin (∠ b' c' a') * dist b' c' / dist a' b'
⊢ ![a, b, c] ∼ ![a', b', c']

Tactic:
]

Goal state after:
V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst✝⁷ : NormedAddCommGroup V₁
inst✝⁶ : NormedAddCommGroup V₂
inst✝⁵ : InnerProductSpace ℝ V₁
inst✝⁴ : InnerProductSpace ℝ V₂
inst✝³ : MetricSpace P₁
inst✝² : MetricSpace P₂
inst✝¹ : NormedAddTorsor V₁ P₁
inst✝ : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h_not_col : ¬Collinear ℝ {a, b, c}
h₁ : ∠ a b c = ∠ a' b' c'
h₂ : ∠ b c a = ∠ b' c' a'
hne_pi_div_two : ∠ a b c ≠ Real.pi / 2 ∨ ∠ b c a ≠ Real.pi / 2
not_all_eq : a' ≠ b' ∨ b' ≠ c' ∨ a' ≠ c'
h_not_col' : ¬Collinear ℝ {a', b', c'}
h_pos1 : 0 < dist a b
h_pos1' : 0 < dist a' b'
h_pos2 : 0 < dist b c
h_pos2' : 0 < dist b' c'
h₃ : ∠ c a b = ∠ c' a' b'
h_sin_ne1 : Real.sin (∠ b c a) ≠ 0
h_sin_ne2 : Real.sin (∠ c a b) ≠ 0
h_sin1 : Real.sin (∠ c a b) = Real.sin (∠ b c a) * dist b c / dist a b
h_sin1' : Real.sin (∠ c' a' b') = Real.sin (∠ b' c' a') * dist b' c' / dist a' b'
⊢ ![a, b, c] ∼ ![a', b', c']
Module: Mathlib.Topology.MetricSpace.Similarity

Goal state before:
pp: case h.«2».«2»
P₁ : Type u_3
P₂ : Type u_4
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
a b c : P₁
a' b' c' : P₂
h_ne : dist a b ≠ 0
h_ne' : dist a' b' ≠ 0
heq1 : dist a b * dist b' c' = dist b c * dist a' b'
heq2 : dist a b * dist c' a' = dist c a * dist a' b'
r : ℝ := dist a b / dist a' b'
hr : r = dist a b / dist a' b'
hr_pos : 0 < r
hij : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨2, ⋯⟩
⊢ 0 = 0

Tactic:
]

Goal state after:
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
n : ℕ
s : Simplex ℝ P (n + 2)
⊢ (affineCombination ℝ univ s.pointsWithCircumcenter)
      ((↑(n + 2 + 1) / ↑(n + 2 - 1)) •
          (centroidWeightsWithCircumcenter univ - circumcenterWeightsWithCircumcenter (n + 2)) +
        circumcenterWeightsWithCircumcenter (n + 2)) =
    (affineCombination ℝ univ s.pointsWithCircumcenter) (mongePointWeightsWithCircumcenter n)

Tactic:
]

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
n : ℕ
s : Simplex ℝ P (n + 2)
⊢ (affineCombination ℝ univ s.pointsWithCircumcenter)
      ((↑(n + 2 + 1) / ↑(n + 2 - 1)) •
          (centroidWeightsWithCircumcenter univ - circumcenterWeightsWithCircumcenter (n + 2)) +
        circumcenterWeightsWithCircumcenter (n + 2)) =
    (affineCombination ℝ univ s.pointsWithCircumcenter) (mongePointWeightsWithCircumcenter n)
Module: Mathlib.Geometry.Euclidean.MongePoint

Goal state before:
pp: case e_6
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)
i : PointsWithCircumcenterIndex (n + 2)
⊢ ↑(n + 2 + 1) / ↑(n + 2 - 1) *
        (centroidWeightsWithCircumcenter univ i - circumcenterWeightsWithCircumcenter (n + 2) i) +
      circumcenterWeightsWithCircumcenter (n + 2) i =
    mongePointWeightsWithCircumcenter n i

Tactic:
]

Goal state after:
case e_6
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)
i : PointsWithCircumcenterIndex (n + 2)
⊢ ↑(n + 2 + 1) / ↑(n + 2 - 1) *
        (centroidWeightsWithCircumcenter univ i - circumcenterWeightsWithCircumcenter (n + 2) i) +
      circumcenterWeightsWithCircumcenter (n + 2) i =
    mongePointWeightsWithCircumcenter n i