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

Goal state before:
pp: case e_a
V : Type u_1
P : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p p₁ p₂ p₃ : P
h₂ : p₁ ≠ p₂
h' : ↑((orthogonalProjection line[ℝ, p₁, p₂]) p) = p₁
hp : p ≠ ↑((orthogonalProjection line[ℝ, p₁, p₂]) p)
hpm : p ∉ line[ℝ, p₁, p₂]
h : inner ℝ (p -ᵥ p₁) (p₃ -ᵥ p₁) = 0
u : V
hu : u ∈ line[ℝ, p₁, p₃].direction
⊢ inner ℝ (p -ᵥ p₁) u = 0

Tactic:
direction_affineSpan,

Goal state after:
case e_a
V : Type u_1
P : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p p₁ p₂ p₃ : P
h₂ : p₁ ≠ p₂
h' : ↑((orthogonalProjection line[ℝ, p₁, p₂]) p) = p₁
hp : p ≠ ↑((orthogonalProjection line[ℝ, p₁, p₂]) p)
hpm : p ∉ line[ℝ, p₁, p₂]
h : inner ℝ (p -ᵥ p₁) (p₃ -ᵥ p₁) = 0
u : V
hu : u ∈ vectorSpan ℝ {p₁, p₃}
⊢ inner ℝ (p -ᵥ p₁) u = 0
Module: Mathlib.Geometry.Euclidean.Sphere.Tangent

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 : Sphere P
p q : P
as : AffineSubspace ℝ P
hp : s.IsTangentAt p as
hq : s.IsTangentAt q as
⊢ p = q

Tactic:
have hqp := hp.inner_left_eq_zero_of_mem hq.mem_space

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 : Sphere P
p q : P
as : AffineSubspace ℝ P
hp : s.IsTangentAt p as
hq : s.IsTangentAt q as
hqp : ⟪q -ᵥ p, p -ᵥ s.center⟫ = 0
⊢ p = q
Module: Mathlib.Geometry.Euclidean.Sphere.OrthRadius

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
inst✝ : FiniteDimensional ℝ V
s : Sphere P
p : P
hp : p ≠ s.center
⊢ 1 + Module.finrank ℝ ↥(mk' p (ℝ ∙ (p -ᵥ s.center))ᗮ).direction = Module.finrank ℝ V

Tactic:
direction_mk'

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
inst✝ : FiniteDimensional ℝ V
s : Sphere P
p : P
hp : p ≠ s.center
⊢ 1 + Module.finrank ℝ ↥(ℝ ∙ (p -ᵥ s.center))ᗮ = Module.finrank ℝ V
Module: Mathlib.Geometry.Euclidean.Sphere.OrthRadius

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
inst✝ : FiniteDimensional ℝ V
s : Sphere P
p : P
hp : p ≠ s.center
⊢ 1 + Module.finrank ℝ ↥(ℝ ∙ (p -ᵥ s.center))ᗮ = Module.finrank ℝ V

Tactic:
convert! (ℝ ∙ (p -ᵥ s.center)).finrank_add_finrank_orthogonal

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
inst✝ : FiniteDimensional ℝ V
s : Sphere P
p : P
hp : p ≠ s.center
⊢ 1 = Module.finrank ℝ ↥(ℝ ∙ (p -ᵥ s.center))
Module: Mathlib.Geometry.Euclidean.Sphere.Tangent

Goal state before:
pp: case inl
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 : Sphere P
as : AffineSubspace ℝ P
p : P
hpar : as ∥ s.orthRadius p
hp : dist p s.center = 0
q : P
hqs : dist q s.center = 0
hqas : q ∈ as
hqo : as ≤ s.orthRadius q
r : ℝ
hr : q -ᵥ s.center = r • (p -ᵥ s.center)
hrad : s.radius = 0
⊢ as = s.orthRadius p ∨ as = s.orthRadius ((Equiv.pointReflection s.center) p)

Tactic:
dist_eq_zero

Goal state after:
case inl
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 : Sphere P
as : AffineSubspace ℝ P
p : P
hpar : as ∥ s.orthRadius p
hp : p = s.center
q : P
hqs : q = s.center
hqas : q ∈ as
hqo : as ≤ s.orthRadius q
r : ℝ
hr : q -ᵥ s.center = r • (p -ᵥ s.center)
hrad : s.radius = 0
⊢ as = s.orthRadius p ∨ as = s.orthRadius ((Equiv.pointReflection s.center) p)
Module: Mathlib.Geometry.Euclidean.Sphere.Tangent

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
inst✝ : Nontrivial V
s₁ s₂ : Sphere P
h : dist s₂.center s₁.center = 0
h₁ : 0 ≤ s₁.radius
h₂ : 0 ≤ s₂.radius
h0 : ¬s₁.center = s₂.center
ha : |s₂.radius - s₁.radius| = s₂.radius - s₁.radius
hr0 : s₂.radius - s₁.radius = 0
⊢ False

Tactic:
dist_eq_zero

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
inst✝ : Nontrivial V
s₁ s₂ : Sphere P
h : s₂.center = s₁.center
h₁ : 0 ≤ s₁.radius
h₂ : 0 ≤ s₂.radius
h0 : ¬s₁.center = s₂.center
ha : |s₂.radius - s₁.radius| = s₂.radius - s₁.radius
hr0 : s₂.radius - s₁.radius = 0
⊢ False
Module: Mathlib.Geometry.Euclidean.Incenter

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
n : ℕ
inst✝¹ : NeZero n
s : Simplex ℝ P n
inst✝ : n.AtLeastTwo
i : Fin (n + 1)
h : (s.height i)⁻¹ + (s.height i)⁻¹ < (s.height i)⁻¹ + ∑ j ∈ {i}ᶜ, (s.height j)⁻¹
⊢ (s.height i)⁻¹ + (s.height i)⁻¹ < ∑ j ∈ {i}, (s.height j)⁻¹ + ∑ j ∈ {i}ᶜ, (s.height j)⁻¹

Tactic:
sum_singleton

Goal state after:
V : Type u_1
P : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
n : ℕ
inst✝¹ : NeZero n
s : Simplex ℝ P n
inst✝ : n.AtLeastTwo
i : Fin (n + 1)
h : (s.height i)⁻¹ + (s.height i)⁻¹ < (s.height i)⁻¹ + ∑ j ∈ {i}ᶜ, (s.height j)⁻¹
⊢ (s.height i)⁻¹ + (s.height i)⁻¹ < (s.height i)⁻¹ + ∑ j ∈ {i}ᶜ, (s.height j)⁻¹
Module: Mathlib.Geometry.Euclidean.Circumcenter

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
s : AffineSubspace ℝ P
inst✝ : s.direction.HasOrthogonalProjection
ps : Set P
p : P
hps : ps ⊆ ↑s
hp : p ∉ s
this : Nonempty ↥s
cc : P
cr : ℝ
hcccru : ∀ (y : Sphere P), y.center ∈ s ∧ ps ⊆ Metric.sphere y.center y.radius → y = { center := cc, radius := cr }
hcc : cc ∈ s
hcr : ps ⊆ Metric.sphere cc cr
x : ℝ := dist cc ↑((orthogonalProjection s) p)
y : ℝ := dist p ↑((orthogonalProjection s) p)
hy0 : y ≠ 0
ycc₂ : ℝ := (x * x + y * y - cr * cr) / (2 * y)
cc₂ : P := (ycc₂ / y) • (p -ᵥ ↑((orthogonalProjection s) p)) +ᵥ cc
cr₂ : ℝ := √(cr * cr + ycc₂ * ycc₂)
hpo : p = 1 • (p -ᵥ ↑((orthogonalProjection s) p)) +ᵥ ↑((orthogonalProjection s) p)
cc₃ : P
cr₃ : ℝ
hcc₃ : cc₃ ∈ affineSpan ℝ (insert p ↑s)
hcr₃ : insert p ps ⊆ Metric.sphere cc₃ cr₃
t₃ : ℝ
hcc₃' : cc ∈ s
hcc₃'' : cc₃ = t₃ • (p -ᵥ ↑((orthogonalProjection s) p)) +ᵥ cc
hcr₃' : ∀ p₁ ∈ ps, dist p₁ ↑⟨cc, hcc₃'⟩ = cr
p0 : P
hp0 : p0 ∈ ps
h' : ↑⟨cc, hcc₃'⟩ = cc
⊢ cr₃ = √(cr * cr + t₃ * y * (t₃ * y))

Tactic:
rw [← dist_of_mem_subset_mk_sphere (Set.mem_insert_of_mem _ hp0) hcr₃, hcc₃'', ←
        mul_self_inj_of_nonneg dist_nonneg (Real.sqrt_nonneg _),
        Real.mul_self_sqrt (add_nonneg (mul_self_nonneg _) (mul_self_nonneg _)),
        dist_sq_eq_dist_orthogonalProjection_sq_add_dist_orthogonalProjection_sq _ (hps hp0),
        orthogonalProjection_vadd_smul_vsub_orthogonalProjection _ _ hcc₃', h',
        dist_of_mem_subset_mk_sphere hp0 hcr, dist_eq_norm_vsub V _ cc, vadd_vsub, norm_smul, ←
        dist_eq_norm_vsub V p, Real.norm_eq_abs, ← mul_assoc, mul_comm _ |t₃|, ← mul_assoc,
        abs_mul_abs_self]

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
s : AffineSubspace ℝ P
inst✝ : s.direction.HasOrthogonalProjection
ps : Set P
p : P
hps : ps ⊆ ↑s
hp : p ∉ s
this : Nonempty ↥s
cc : P
cr : ℝ
hcccru : ∀ (y : Sphere P), y.center ∈ s ∧ ps ⊆ Metric.sphere y.center y.radius → y = { center := cc, radius := cr }
hcc : cc ∈ s
hcr : ps ⊆ Metric.sphere cc cr
x : ℝ := dist cc ↑((orthogonalProjection s) p)
y : ℝ := dist p ↑((orthogonalProjection s) p)
hy0 : y ≠ 0
ycc₂ : ℝ := (x * x + y * y - cr * cr) / (2 * y)
cc₂ : P := (ycc₂ / y) • (p -ᵥ ↑((orthogonalProjection s) p)) +ᵥ cc
cr₂ : ℝ := √(cr * cr + ycc₂ * ycc₂)
hpo : p = 1 • (p -ᵥ ↑((orthogonalProjection s) p)) +ᵥ ↑((orthogonalProjection s) p)
cc₃ : P
cr₃ : ℝ
hcc₃ : cc₃ ∈ affineSpan ℝ (insert p ↑s)
hcr₃ : insert p ps ⊆ Metric.sphere cc₃ cr₃
t₃ : ℝ
hcc₃' : cc ∈ s
hcc₃'' : cc₃ = t₃ • (p -ᵥ ↑((orthogonalProjection s) p)) +ᵥ cc
hcr₃' : ∀ p₁ ∈ ps, dist p₁ ↑⟨cc, hcc₃'⟩ = cr
p0 : P
hp0 : p0 ∈ ps
h' : ↑⟨cc, hcc₃'⟩ = cc
⊢ cr * cr + t₃ * t₃ * dist p ↑((orthogonalProjection s) p) * dist p ↑((orthogonalProjection s) p) =
    cr * cr + t₃ * y * (t₃ * y)
Module: Mathlib.Geometry.Euclidean.Circumcenter

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 + 1)
h : 0 = s.circumradius
hr : ∀ (i : Fin (n + 1 + 1)), dist (s.points i) s.circumcenter = 0
⊢ False

Tactic:
dist_eq_zero

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 + 1)
h : 0 = s.circumradius
hr : ∀ (i : Fin (n + 1 + 1)), s.points i = s.circumcenter
⊢ False
Module: Mathlib.Geometry.Euclidean.Circumcenter

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 : Simplex ℝ P 0
i : Fin 1
⊢ s.circumcenter = s.points i

Tactic:
have h := s.circumcenter_mem_affineSpan

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 : Simplex ℝ P 0
i : Fin 1
h : s.circumcenter ∈ affineSpan ℝ (Set.range s.points)
⊢ s.circumcenter = s.points i
Module: Mathlib.Geometry.Euclidean.Angle.Sphere

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 i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
S : AffineSubspace ℝ P := affineSpan ℝ (Set.range t.points)
hS : S = affineSpan ℝ (Set.range t.points)
t' : Triangle ℝ ↥S := Simplex.restrict t S ⋯
⊢ finrank ℝ ↥(affineSpan ℝ (Set.range t.points)).direction = 2

Tactic:
direction_affineSpan,

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 i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
S : AffineSubspace ℝ P := affineSpan ℝ (Set.range t.points)
hS : S = affineSpan ℝ (Set.range t.points)
t' : Triangle ℝ ↥S := Simplex.restrict t S ⋯
⊢ finrank ℝ ↥(vectorSpan ℝ (Set.range t.points)) = 2
Module: Mathlib.Geometry.Euclidean.Incenter

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
hf : Fact (Module.finrank ℝ V = n)
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i : Fin (n + 1)
⊢ Module.finrank ℝ ↥(affineSpan ℝ (Set.range (s.faceOpposite i).points)).direction + 1 = Module.finrank ℝ V

Tactic:
direction_affineSpan,

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
hf : Fact (Module.finrank ℝ V = n)
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i : Fin (n + 1)
⊢ Module.finrank ℝ ↥(vectorSpan ℝ (Set.range (s.faceOpposite i).points)) + 1 = Module.finrank ℝ V
Module: Mathlib.Geometry.Euclidean.Circumcenter

Goal state before:
pp: n : ℕ
i₁ i₂ : Fin (n + 1)
h : i₁ ≠ i₂
⊢ #{x | x = i₁ ∨ x = i₂} • 1 + #{x | ¬(x = i₁ ∨ x = i₂)} • 0 + -1 = 1

Tactic:
filter_or,

Goal state after:
n : ℕ
i₁ i₂ : Fin (n + 1)
h : i₁ ≠ i₂
⊢ #({a | a = i₁} ∪ {a | a = i₂}) • 1 + #{x | ¬(x = i₁ ∨ x = i₂)} • 0 + -1 = 1
Module: Mathlib.Geometry.Euclidean.Incenter

Goal state before:
pp: case refine_2
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
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hne : i ≠ j
r : ℝ
hr : r ∈ Set.Icc 0 1
t : ℝ
ht : t ∈ Set.Icc 0 1
⊢ Continuous ⇑(AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i))

Tactic:
rw [← ContinuousAffineMap.lineMap_toAffineMap]

Goal state after:
case refine_2
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
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hne : i ≠ j
r : ℝ
hr : r ∈ Set.Icc 0 1
t : ℝ
ht : t ∈ Set.Icc 0 1
⊢ Continuous ⇑↑(ContinuousAffineMap.lineMap (s.excenter signs) (s.touchpoint signs i))
Module: Mathlib.Geometry.Euclidean.Incenter

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
n : ℕ
inst✝¹ : NeZero n
s : Simplex ℝ P n
inst✝ : n.AtLeastTwo
i j : Fin (n + 1)
hne : i ≠ j
⊢ s.touchpointWeights {i} j i < 0

Tactic:
by
  simpa [sign_eq_neg_one_iff] using s.sign_touchpointWeights_singleton_neg hne

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

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 i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
⊢ line[ℝ, t.points i₂, t.points i₃].SSameSide (t.points i₁) (Simplex.incenter t)

Tactic:
by
  convert! t.sSameSide_point_incenter i₁
  simp
  grind

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 ∈ (affineSpan ℝ (Set.range (Simplex.faceOpposite s i).points)).directionᗮ

Tactic:
direction_affineSpan,

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 (Set.range (Simplex.faceOpposite s i).points))ᗮ
Module: Mathlib.Geometry.Euclidean.MongePoint

Goal state before:
pp: n : ℕ
⊢ ∑ i, mongePointWeightsWithCircumcenter n i = 1

Tactic:
simp_rw [sum_pointsWithCircumcenter, mongePointWeightsWithCircumcenter, sum_const, card_fin,
    nsmul_eq_mul]

Goal state after:
n : ℕ
⊢ ↑(n + 2 + 1) * (↑(n + 1))⁻¹ + -2 / ↑(n + 1) = 1
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
hr : dist (s.points i₁) (s.points i₂) = 0
⊢ False

Tactic:
dist_eq_zero

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
hr : s.points i₁ = s.points i₂
⊢ False
Module: Mathlib.Geometry.Euclidean.MongePoint

Goal state before:
pp: case pointIndex
n : ℕ
i₁ i₂ : Fin (n + 3)
h : i₁ ≠ i₂
i : Fin (n + 2 + 1)
⊢ mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂ (pointIndex i) =
    (↑(n + 1))⁻¹ - if i ∈ {i₁, i₂}ᶜ then (↑(#{i₁, i₂}ᶜ))⁻¹ else 0

Tactic:
mongePointVSubFaceCentroidWeightsWithCircumcenter

Goal state after:
case pointIndex
n : ℕ
i₁ i₂ : Fin (n + 3)
h : i₁ ≠ i₂
i : Fin (n + 2 + 1)
⊢ (if i = i₁ ∨ i = i₂ then (↑(n + 1))⁻¹ else 0) = (↑(n + 1))⁻¹ - if i ∈ {i₁, i₂}ᶜ then (↑(#{i₁, i₂}ᶜ))⁻¹ else 0
Module: Mathlib.Geometry.Euclidean.MongePoint

Goal state before:
pp: case left
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, (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i = 0
fs : Finset (Fin (n + 3)) := ⋯
i : Fin (n + 3)
hi : i ∉ fs
⊢ i ≠ i₁

Tactic:
intro hj

Goal state after:
case left
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, (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i = 0
fs : Finset (Fin (n + 3)) := {i₁, i₂}
i : Fin (n + 3)
hi : i ∉ fs
hj : i = i₁
⊢ False
Module: Mathlib.Geometry.Euclidean.MongePoint

Goal state before:
pp: case right
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, (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i = 0
fs : Finset (Fin (n + 3)) := ⋯
i : Fin (n + 3)
hi : i ∉ fs
⊢ i ≠ i₂

Tactic:
intro hj

Goal state after:
case right
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, (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i = 0
fs : Finset (Fin (n + 3)) := {i₁, i₂}
i : Fin (n + 3)
hi : i ∉ fs
hj : i = i₂
⊢ False
Module: Mathlib.Geometry.Euclidean.Sphere.Power

Goal state before:
pp: V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ p₄ p : P
h : dist p₁ p * dist p₂ p = dist p₃ p * dist p₄ p
hp₁p₂ : ∠ p₁ p p₂ = π
hp₃p₄ : ∠ p₃ p p₄ = π
hn : ¬Collinear ℝ {p₁, p, p₃}
hp₁p₂_sbtw : Sbtw ℝ p₁ p p₂
hp₃p₄_sbtw : Sbtw ℝ p₃ p p₄
hindep : AffineIndependent ℝ ![p₁, p, p₃]
t : Affine.Triangle ℝ P := { points := ![p₁, p, p₃], independent := hindep }
ht : t = { points := ![p₁, p, p₃], independent := hindep }
S : AffineSubspace ℝ P := affineSpan ℝ (Set.range t.points)
hS : S = affineSpan ℝ (Set.range t.points)
hp₂ : p₂ ∈ S
hp₄ : p₄ ∈ S
s_isom : ↥S →ᵃⁱ[ℝ] P := S.subtypeₐᵢ
p₁' : ↥S := ⟨p₁, ⋯⟩
p' : ↥S := ⟨p, ⋯⟩
p₃' : ↥S := ⟨p₃, ⋯⟩
p₂' : ↥S := ⟨p₂, hp₂⟩
p₄' : ↥S := ⟨p₄, hp₄⟩
h_dist' : dist p₁' p' * dist p₂' p' = dist p₃' p' * dist p₄' p'
hp₁'p₂' : ∠ p₁' p' p₂' = π
hp₃'p₄' : ∠ p₃' p' p₄' = π
⊢ finrank ℝ ↥(affineSpan ℝ (Set.range t.points)).direction = 2

Tactic:
direction_affineSpan,

Goal state after:
V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ p₄ p : P
h : dist p₁ p * dist p₂ p = dist p₃ p * dist p₄ p
hp₁p₂ : ∠ p₁ p p₂ = π
hp₃p₄ : ∠ p₃ p p₄ = π
hn : ¬Collinear ℝ {p₁, p, p₃}
hp₁p₂_sbtw : Sbtw ℝ p₁ p p₂
hp₃p₄_sbtw : Sbtw ℝ p₃ p p₄
hindep : AffineIndependent ℝ ![p₁, p, p₃]
t : Affine.Triangle ℝ P := { points := ![p₁, p, p₃], independent := hindep }
ht : t = { points := ![p₁, p, p₃], independent := hindep }
S : AffineSubspace ℝ P := affineSpan ℝ (Set.range t.points)
hS : S = affineSpan ℝ (Set.range t.points)
hp₂ : p₂ ∈ S
hp₄ : p₄ ∈ S
s_isom : ↥S →ᵃⁱ[ℝ] P := S.subtypeₐᵢ
p₁' : ↥S := ⟨p₁, ⋯⟩
p' : ↥S := ⟨p, ⋯⟩
p₃' : ↥S := ⟨p₃, ⋯⟩
p₂' : ↥S := ⟨p₂, hp₂⟩
p₄' : ↥S := ⟨p₄, hp₄⟩
h_dist' : dist p₁' p' * dist p₂' p' = dist p₃' p' * dist p₄' p'
hp₁'p₂' : ∠ p₁' p' p₂' = π
hp₃'p₄' : ∠ p₃' p' p₄' = π
⊢ finrank ℝ ↥(vectorSpan ℝ (Set.range t.points)) = 2
Module: Mathlib.Geometry.Euclidean.Similarity

Goal state before:
pp: case pos.inl
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
r : ℝ
hr_pos : 0 < r
hdist :
  ∀ (i₁ i₂ : Fin (Nat.succ 0).succ.succ),
    dist (![a, b, c] i₁) (![a, b, c] i₂) = r * dist (![a', b', c'] i₁) (![a', b', c'] i₂)
h_ab : dist a b = r * dist a' b'
h_cb : dist c b = r * dist c' b'
h_ac : dist a c = r * dist a' c'
h_cos : dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c') = dist a' b' * dist c' b' * 2 * Real.cos (∠ a b c)
h_cos' : dist a' c' ^ 2 = dist a' b' ^ 2 + dist c' b' ^ 2 - dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c')
h1 : dist a' b' = 0
h_dist_ab : dist a b = 0
⊢ ∠ a b c = ∠ a' b' c'

Tactic:
dist_eq_zero

Goal state after:
case pos.inl
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
r : ℝ
hr_pos : 0 < r
hdist :
  ∀ (i₁ i₂ : Fin (Nat.succ 0).succ.succ),
    dist (![a, b, c] i₁) (![a, b, c] i₂) = r * dist (![a', b', c'] i₁) (![a', b', c'] i₂)
h_ab : dist a b = r * dist a' b'
h_cb : dist c b = r * dist c' b'
h_ac : dist a c = r * dist a' c'
h_cos : dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c') = dist a' b' * dist c' b' * 2 * Real.cos (∠ a b c)
h_cos' : dist a' c' ^ 2 = dist a' b' ^ 2 + dist c' b' ^ 2 - dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c')
h1 : a' = b'
h_dist_ab : a = b
⊢ ∠ a b c = ∠ a' b' c'
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 i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
hs : {i₂, i₃}ᶜ = {i₁}
he : {i₁}ᶜ = {i₂, i₃}
⊢ mk' (t.points i₁) (affineSpan ℝ (t.points '' {i₁}ᶜ)).directionᗮ ⊓ affineSpan ℝ (Set.range t.points) =
    mk' (Finset.centroid ℝ {i₂, i₃}ᶜ t.points) (ℝ ∙ (t.points i₂ -ᵥ t.points i₃))ᗮ ⊓ affineSpan ℝ (Set.range t.points)

Tactic:
direction_affineSpan,

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 i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
hs : {i₂, i₃}ᶜ = {i₁}
he : {i₁}ᶜ = {i₂, i₃}
⊢ mk' (t.points i₁) (vectorSpan ℝ (t.points '' {i₁}ᶜ))ᗮ ⊓ affineSpan ℝ (Set.range t.points) =
    mk' (Finset.centroid ℝ {i₂, i₃}ᶜ t.points) (ℝ ∙ (t.points i₂ -ᵥ t.points i₃))ᗮ ⊓ affineSpan ℝ (Set.range t.points)