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

Goal state before:
pp: V : Type u_3
P : Type u_4
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
s : Sphere P
p₁ p₂ p₃ : P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
hp₃ : p₃ ∈ s
hp₂p₁ : p₂ ≠ p₁
hp₂p₃ : p₂ ≠ p₃
hp₁p₃ : p₁ ≠ p₃
⊢ 2 • ∡ p₃ p₁ s.center + 2 • ∡ p₁ p₂ p₃ = ↑π

Tactic:
← oangle_center_eq_two_zsmul_oangle hp₁ hp₂ hp₃ hp₂p₁ hp₂p₃,

Goal state after:
V : Type u_3
P : Type u_4
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
s : Sphere P
p₁ p₂ p₃ : P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
hp₃ : p₃ ∈ s
hp₂p₁ : p₂ ≠ p₁
hp₂p₃ : p₂ ≠ p₃
hp₁p₃ : p₁ ≠ p₃
⊢ 2 • ∡ p₃ p₁ s.center + ∡ p₁ s.center p₃ = ↑π
Module: Mathlib.Geometry.Euclidean.Angle.Sphere

Goal state before:
pp: V : Type u_3
P : Type u_4
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
s : Sphere P
p₁ p₂ : P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
h : p₁ ≠ p₂
⊢ ((∡ p₂ p₁ s.center).tan / 2) • («o».rotation ↑(π / 2)) (p₂ -ᵥ p₁) +ᵥ midpoint ℝ p₁ p₂ = s.center

Tactic:
by
  obtain ⟨r, hr⟩ := (dist_eq_iff_eq_smul_rotation_pi_div_two_vadd_midpoint h).1
    (dist_center_eq_dist_center_of_mem_sphere hp₁ hp₂)
  rw [← hr, ← oangle_midpoint_rev_left, oangle, vadd_vsub_assoc]
  nth_rw 1 [show p₂ -ᵥ p₁ = (2 : ℝ) • (midpoint ℝ p₁ p₂ -ᵥ p₁) by simp]
  rw [map_smul, smul_smul, add_comm, o.tan_oangle_add_right_smul_rotation_pi_div_two,
    mul_div_cancel_right₀ _ (two_ne_zero' ℝ)]
  simpa using h.symm

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

Goal state before:
pp: case intro.zero
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
ι : Type u_3
hne : Nonempty ι
inst✝ : Finite ι
p : ι → P
ha : AffineIndependent ℝ p
val✝ : Fintype ι
hn : Fintype.card ι = 0
⊢ ∃! cs, cs.center ∈ affineSpan ℝ (Set.range p) ∧ Set.range p ⊆ Metric.sphere cs.center cs.radius

Tactic:
exfalso
    have h := Fintype.card_pos_iff.2 hne
    lia

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

Goal state before:
pp: case pos
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
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hij : i = j
⊢ s.excenter signs ∉ line[ℝ, s.points i, s.points j]

pp: case neg
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
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hij : ¬i = j
⊢ s.excenter signs ∉ line[ℝ, s.points i, s.points j]

Tactic:
· simp only [hij, Set.mem_singleton_iff, Set.insert_eq_of_mem,
      AffineSubspace.mem_affineSpan_singleton]
    exact h.excenter_ne_point j

Goal state after:
case neg
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
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hij : ¬i = j
⊢ s.excenter signs ∉ line[ℝ, s.points i, s.points j]
Module: Mathlib.Geometry.Euclidean.Circumcenter

Goal state before:
pp: case intro.succ.succ
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
ι : Type u_3
hne : Nonempty ι
inst✝ : Finite ι
p : ι → P
ha : AffineIndependent ℝ p
val✝ : Fintype ι
m : ℕ
hm :
  ∀ {ι : Type u_3} [hne : Nonempty ι] [Finite ι] {p : ι → P},
    AffineIndependent ℝ p →
      ∀ (val : Fintype ι),
        Fintype.card ι = m + 1 →
          ∃! cs, cs.center ∈ affineSpan ℝ (Set.range p) ∧ Set.range p ⊆ Metric.sphere cs.center cs.radius
hn : Fintype.card ι = m + 1 + 1
⊢ ∃! cs, cs.center ∈ affineSpan ℝ (Set.range p) ∧ Set.range p ⊆ Metric.sphere cs.center cs.radius

Tactic:
have i := hne.some
      let ι2 := { x // x ≠ i }
      classical
      have hc : Fintype.card ι2 = m + 1 := by
        rw [Fintype.card_of_subtype {x | x ≠ i}]
        · rw [Finset.filter_not, Finset.filter_eq' _ i, if_pos (Finset.mem_univ _),
            Finset.card_sdiff, Finset.card_univ, hn]
          simp
        · simp
      have : Nonempty ι2 := Fintype.card_pos_iff.1 (hc.symm ▸ Nat.zero_lt_succ _)
      have ha2 : AffineIndependent ℝ fun i2 : ι2 => p i2 := ha.subtype _
      replace hm := hm ha2 _ hc
      have hr : Set.range p = insert (p i) (Set.range fun i2 : ι2 => p i2) := by
        change _ = insert _ (Set.range fun i2 : { x | x ≠ i } => p i2)
        rw [← Set.image_eq_range, ← Set.image_univ, ← Set.image_insert_eq]
        congr with j
        simp [Classical.em]
      rw [hr, ← affineSpan_insert_affineSpan]
      refine existsUnique_dist_eq_of_insert (Set.range_nonempty _) (subset_affineSpan ℝ _) ?_ hm
      convert! ha.notMem_affineSpan_sdiff i Set.univ
      change (Set.range fun i2 : { x | x ≠ i } => p i2) = _
      rw [← Set.image_eq_range]
      congr 1 with j
      simp

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

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hy : y ≠ 0
hx : x ≠ 0
h : ↑(angle x y) = ↑(angle x (x + y) + angle y (x + y))
n : ℤ
this : -1 < n
hn : 1 ≤ n
hxy : x + y ≠ 0
r₁ : ℝ
r₁_pos : r₁ < 0
hr₁ : x = r₁ • (x + y)
r₂ : ℝ
r₂_pos : r₂ < 0
hr₂ : y = r₂ • (x + y)
⊢ (r₁ + r₂) • (x + y) - 1 • (x + y) = 0

Tactic:
add_smul,

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hy : y ≠ 0
hx : x ≠ 0
h : ↑(angle x y) = ↑(angle x (x + y) + angle y (x + y))
n : ℤ
this : -1 < n
hn : 1 ≤ n
hxy : x + y ≠ 0
r₁ : ℝ
r₁_pos : r₁ < 0
hr₁ : x = r₁ • (x + y)
r₂ : ℝ
r₂_pos : r₂ < 0
hr₂ : y = r₂ • (x + y)
⊢ r₁ • (x + y) + r₂ • (x + y) - 1 • (x + y) = 0
Module: Mathlib.Geometry.Euclidean.Angle.Sphere

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
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
t : Triangle ℝ P
p : P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h : 2 • ∡ (t.points i₁) p (t.points i₃) = 2 • ∡ (t.points i₁) (t.points i₂) (t.points i₃)
t'p : Fin 3 → P := Function.update t.points i₂ p
h₁ : t'p i₁ = t.points i₁
h₂ : t'p i₂ = p
h₃ : t'p i₃ = t.points i₃
⊢ p ∈ Simplex.circumsphere t

Tactic:
have ha : AffineIndependent ℝ t'p := by
    rw [affineIndependent_iff_not_collinear_of_ne h₁₂ h₁₃ h₂₃, h₁, h₂, h₃,
      collinear_iff_of_two_zsmul_oangle_eq h, ←
      affineIndependent_iff_not_collinear_of_ne h₁₂ h₁₃ h₂₃]
    exact t.independent

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
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
t : Triangle ℝ P
p : P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h : 2 • ∡ (t.points i₁) p (t.points i₃) = 2 • ∡ (t.points i₁) (t.points i₂) (t.points i₃)
t'p : Fin 3 → P := Function.update t.points i₂ p
h₁ : t'p i₁ = t.points i₁
h₂ : t'p i₂ = p
h₃ : t'p i₃ = t.points i₃
ha : AffineIndependent ℝ t'p
⊢ p ∈ Simplex.circumsphere t
Module: Mathlib.Geometry.Euclidean.Circumcenter

Goal state before:
pp: case refine_1
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
m n : ℕ
s : Simplex ℝ P m
e : Fin (m + 1) ≃ Fin (n + 1)
⊢ (s.reindex e).circumsphere.center ∈ affineSpan ℝ (Set.range (s.reindex e).points)

pp: case refine_2
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
m n : ℕ
s : Simplex ℝ P m
e : Fin (m + 1) ≃ Fin (n + 1)
⊢ Set.range (s.reindex e).points ⊆ Metric.sphere (s.reindex e).circumsphere.center (s.reindex e).circumsphere.radius

Tactic:
· exact (s.reindex e).circumsphere_unique_dist_eq.1.1

Goal state after:
case refine_2
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
m n : ℕ
s : Simplex ℝ P m
e : Fin (m + 1) ≃ Fin (n + 1)
⊢ Set.range (s.reindex e).points ⊆ Metric.sphere (s.reindex e).circumsphere.center (s.reindex e).circumsphere.radius
Module: Mathlib.Geometry.Euclidean.Triangle

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
inst✝¹ : Module.Oriented ℝ V (Fin 2)
inst✝ : Fact (Module.finrank ℝ V = 2)
p₁ p₂ p₃ : P
h : 2 • ∡ p₁ p₂ p₃ = 2 • ∡ p₂ p₃ p₁
h0 : ∡ p₃ p₁ p₂ ≠ 0
hpi : ∡ p₃ p₁ p₂ ≠ ↑π
⊢ 2 • o.oangle (-(p₂ -ᵥ p₁)) (-(p₂ -ᵥ p₁ - (p₃ -ᵥ p₁))) = 2 • o.oangle (-(p₃ -ᵥ p₁ - (p₂ -ᵥ p₁))) (-(p₃ -ᵥ p₁))

Tactic:
oangle

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
inst✝¹ : Module.Oriented ℝ V (Fin 2)
inst✝ : Fact (Module.finrank ℝ V = 2)
p₁ p₂ p₃ : P
h : 2 • o.oangle (p₁ -ᵥ p₂) (p₃ -ᵥ p₂) = 2 • o.oangle (p₂ -ᵥ p₃) (p₁ -ᵥ p₃)
h0 : ∡ p₃ p₁ p₂ ≠ 0
hpi : ∡ p₃ p₁ p₂ ≠ ↑π
⊢ 2 • o.oangle (-(p₂ -ᵥ p₁)) (-(p₂ -ᵥ p₁ - (p₃ -ᵥ p₁))) = 2 • o.oangle (-(p₃ -ᵥ p₁ - (p₂ -ᵥ p₁))) (-(p₃ -ᵥ p₁))
Module: Mathlib.Geometry.Euclidean.Triangle

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
p₁ p₂ p₃ : P
h : p₂ ≠ p₁
⊢ InnerProductGeometry.angle (p₃ -ᵥ p₁) (p₂ -ᵥ p₁) + InnerProductGeometry.angle (p₃ -ᵥ p₁) (p₃ -ᵥ p₁ - (p₂ -ᵥ p₁)) +
      InnerProductGeometry.angle (p₂ -ᵥ p₁) (p₂ -ᵥ p₁ - (p₃ -ᵥ p₁)) =
    π

Tactic:
exact angle_add_angle_sub_add_angle_sub_eq_pi _ fun he => h (vsub_eq_zero_iff_eq.1 he)

Goal state after:
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
i₁ i₂ : Fin (n + 1)
h : i₁ ≠ i₂
hc : #{i₁, i₂} = 2
W : AffineSubspace ℝ P := affineSpan ℝ (s.points '' {i₁, i₂})
h_faces : ↑((orthogonalProjection W) s.circumcenter) = ↑((s.face hc).orthogonalProjectionSpan s.circumcenter)
⊢ ∑ x,
        ((centroidWeightsWithCircumcenter {i₁, i₂} (pointIndex x) -
                circumcenterWeightsWithCircumcenter n (pointIndex x) +
              centroidWeightsWithCircumcenter {i₁, i₂} (pointIndex x)) •
            (s.pointsWithCircumcenter (pointIndex x) -ᵥ Classical.choice ⋯) -
          reflectionCircumcenterWeightsWithCircumcenter i₁ i₂ (pointIndex x) •
            (s.pointsWithCircumcenter (pointIndex x) -ᵥ Classical.choice ⋯)) +
      ((centroidWeightsWithCircumcenter {i₁, i₂} circumcenterIndex -
              circumcenterWeightsWithCircumcenter n circumcenterIndex +
            centroidWeightsWithCircumcenter {i₁, i₂} circumcenterIndex) •
          (s.pointsWithCircumcenter circumcenterIndex -ᵥ Classical.choice ⋯) -
        reflectionCircumcenterWeightsWithCircumcenter i₁ i₂ circumcenterIndex •
          (s.pointsWithCircumcenter circumcenterIndex -ᵥ Classical.choice ⋯)) =
    0

Tactic:
add_smul,

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 : Fin (n + 1)
h : i₁ ≠ i₂
hc : #{i₁, i₂} = 2
W : AffineSubspace ℝ P := affineSpan ℝ (s.points '' {i₁, i₂})
h_faces : ↑((orthogonalProjection W) s.circumcenter) = ↑((s.face hc).orthogonalProjectionSpan s.circumcenter)
⊢ ∑ x,
        ((centroidWeightsWithCircumcenter {i₁, i₂} (pointIndex x) -
                circumcenterWeightsWithCircumcenter n (pointIndex x)) •
              (s.pointsWithCircumcenter (pointIndex x) -ᵥ Classical.choice ⋯) +
            centroidWeightsWithCircumcenter {i₁, i₂} (pointIndex x) •
              (s.pointsWithCircumcenter (pointIndex x) -ᵥ Classical.choice ⋯) -
          reflectionCircumcenterWeightsWithCircumcenter i₁ i₂ (pointIndex x) •
            (s.pointsWithCircumcenter (pointIndex x) -ᵥ Classical.choice ⋯)) +
      ((centroidWeightsWithCircumcenter {i₁, i₂} circumcenterIndex -
              circumcenterWeightsWithCircumcenter n circumcenterIndex) •
            (s.pointsWithCircumcenter circumcenterIndex -ᵥ Classical.choice ⋯) +
          centroidWeightsWithCircumcenter {i₁, i₂} circumcenterIndex •
            (s.pointsWithCircumcenter circumcenterIndex -ᵥ Classical.choice ⋯) -
        reflectionCircumcenterWeightsWithCircumcenter i₁ i₂ circumcenterIndex •
          (s.pointsWithCircumcenter circumcenterIndex -ᵥ Classical.choice ⋯)) =
    0
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
ps : Set P
n : ℕ
inst✝ : FiniteDimensional ℝ V
hd : finrank ℝ V = n
hc : Cospherical ps
sx₁ sx₂ : Simplex ℝ P n
hsx₁ : Set.range sx₁.points ⊆ ps
hsx₂ : Set.range sx₂.points ⊆ ps
⊢ sx₁.circumsphere = sx₂.circumsphere

Tactic:
rcases exists_circumsphere_eq_of_cospherical hd hc with ⟨r, hr⟩
  rw [hr sx₁ hsx₁, hr sx₂ hsx₂]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.TriangleInequality

Goal state before:
pp: case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y z : V
hy : y ≠ 0
h : angle x z = angle x y + angle y z
hxz₁ : ¬angle x z = π
hxz₂ : ¬angle x z = 0
h_sin_xz : Real.sin (angle x z) ≠ 0
hx : ¬x = 0
hz : ¬z = 0
⊢ y ∈ Submodule.span ℝ≥0 {x, z}

Tactic:
have hnneg₁ : 0 ≤ Real.sin (angle y z) / ‖x‖ := by
      grind [sin_angle_nonneg, norm_nonneg, div_nonneg_iff, mul_nonneg_iff]

Goal state after:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y z : V
hy : y ≠ 0
h : angle x z = angle x y + angle y z
hxz₁ : ¬angle x z = π
hxz₂ : ¬angle x z = 0
h_sin_xz : Real.sin (angle x z) ≠ 0
hx : ¬x = 0
hz : ¬z = 0
hnneg₁ : 0 ≤ Real.sin (angle y z) / ‖x‖
⊢ y ∈ Submodule.span ℝ≥0 {x, z}
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)
⊢ ‖↑n • (s.faceOppositeCentroid i -ᵥ s.circumcenter) - (↑n + 1) • (s.centroid -ᵥ s.circumcenter)‖ =
    ‖s.circumcenter -ᵥ s.points i‖

Tactic:
add_smul,

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)
⊢ ‖↑n • (s.faceOppositeCentroid i -ᵥ s.circumcenter) -
        (↑n • (s.centroid -ᵥ s.circumcenter) + 1 • (s.centroid -ᵥ s.circumcenter))‖ =
    ‖s.circumcenter -ᵥ s.points i‖
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
hn : NeZero 1
s : Simplex ℝ P 1
i : Fin (1 + 1)
this : 2⁻¹ • (s.faceOppositeCentroid i -ᵥ s.centroid) + 2⁻¹ • (s.points i -ᵥ s.centroid) = 0
⊢ ⅟2 • (s.faceOppositeCentroid i -ᵥ s.circumcenter) +
      ⅟2 •
        (((↑1)⁻¹ • (s.points i -ᵥ ((↑(1 + 1) / ↑(1 - 1)) • (s.centroid -ᵥ s.circumcenter) +ᵥ s.circumcenter)) +ᵥ
            (↑(1 + 1) / ↑(1 - 1)) • (s.centroid -ᵥ s.circumcenter) +ᵥ s.circumcenter) -ᵥ
          s.circumcenter) =
    ((↑1 + 1) / ↑1) • (s.centroid -ᵥ s.circumcenter)

Tactic:
by
      simpa [circumcenter_eq_centroid, centroid]

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)
hn1 : ¬n = 1
hltn : 1 < n
hnsub1 : ↑(n - 1) = ↑n - 1
⊢ (↑n)⁻¹ • (s.centroid -ᵥ s.circumcenter) + (s.centroid -ᵥ s.circumcenter) =
    (1 / ↑n + 1) • (s.centroid -ᵥ s.circumcenter)

Tactic:
add_smul,

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)
hn1 : ¬n = 1
hltn : 1 < n
hnsub1 : ↑(n - 1) = ↑n - 1
⊢ (↑n)⁻¹ • (s.centroid -ᵥ s.circumcenter) + (s.centroid -ᵥ s.circumcenter) =
    (1 / ↑n) • (s.centroid -ᵥ s.circumcenter) + 1 • (s.centroid -ᵥ s.circumcenter)
Module: Mathlib.Geometry.Euclidean.Sphere.Power

Goal state before:
pp: V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
P : Type u_2
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ p₄ p : P
h : dist p₁ p * dist p₂ p = dist p₃ p * dist p₄ p
hp₁p₂ : Sbtw ℝ p₁ p p₂
hp₃p₄ : Sbtw ℝ p₃ p p₄
hn : ¬Collinear ℝ {p₁, p, p₃}
h_angle_eq : ∠ p₁ p p₄ = ∠ p₃ p p₂
hcol_p₁pp₂ : Collinear ℝ {p₁, p, p₂}
hcol_p₃pp₄ : Collinear ℝ {p₃, p, p₄}
h_notcol_p₁p₂p₃ : ¬Collinear ℝ {p₁, p₂, p₃}
⊢ Cospherical {p₁, p₂, p₄, p₃}

Tactic:
apply cospherical_of_two_zsmul_oangle_eq_of_not_collinear ?_ h_notcol_p₁p₂p₃

Goal state after:
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
P : Type u_2
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ p₄ p : P
h : dist p₁ p * dist p₂ p = dist p₃ p * dist p₄ p
hp₁p₂ : Sbtw ℝ p₁ p p₂
hp₃p₄ : Sbtw ℝ p₃ p p₄
hn : ¬Collinear ℝ {p₁, p, p₃}
h_angle_eq : ∠ p₁ p p₄ = ∠ p₃ p p₂
hcol_p₁pp₂ : Collinear ℝ {p₁, p, p₂}
hcol_p₃pp₄ : Collinear ℝ {p₃, p, p₄}
h_notcol_p₁p₂p₃ : ¬Collinear ℝ {p₁, p₂, p₃}
⊢ 2 • ∡ p₁ p₂ p₃ = 2 • ∡ p₁ p₄ p₃
Module: Mathlib.Geometry.Euclidean.Sphere.Ptolemy

Goal state before:
pp: case h
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
a b c d p : P
h : Cospherical {a, b, c, d}
hapc : ∠ a p c = π
hbpd : ∠ b p d = π
h' : Cospherical {a, c, b, d}
hmul : dist a p * dist c p = dist b p * dist d p
hbp : dist b p ≠ 0
⊢ ∠ c p d = ∠ b p a

Tactic:
angle_eq_angle_of_angle_eq_pi_of_angle_eq_pi hbpd hapc,

Goal state after:
case h
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
a b c d p : P
h : Cospherical {a, b, c, d}
hapc : ∠ a p c = π
hbpd : ∠ b p d = π
h' : Cospherical {a, c, b, d}
hmul : dist a p * dist c p = dist b p * dist d p
hbp : dist b p ≠ 0
⊢ ∠ c p d = ∠ d p c
Module: Mathlib.Geometry.Euclidean.Sphere.SecondInter

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
s : Sphere P
p : P
v : V
hp : p -ᵥ p = (-2 * ⟪v, p -ᵥ s.center⟫ / ⟪v, v⟫) • v
hv : ¬v = 0
⊢ ⟪v, p -ᵥ s.center⟫ = 0

Tactic:
vsub_self,

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
s : Sphere P
p : P
v : V
hp : 0 = (-2 * ⟪v, p -ᵥ s.center⟫ / ⟪v, v⟫) • v
hv : ¬v = 0
⊢ ⟪v, p -ᵥ s.center⟫ = 0
Module: Mathlib.Geometry.Euclidean.Sphere.Power

Goal state before:
pp: P : Type u_2
inst✝ : MetricSpace P
s : Sphere P
p : P
hr : 0 ≤ s.radius
⊢ dist p s.center ^ 2 = s.radius ^ 2 ↔ dist p s.center = s.radius

Tactic:
pow_left_inj₀ dist_nonneg hr two_ne_zero

Goal state after:
P : Type u_2
inst✝ : MetricSpace P
s : Sphere P
p : P
hr : 0 ≤ s.radius
⊢ dist p s.center = s.radius ↔ dist p s.center = s.radius
Module: Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
u : Opens M
⊢ ModelWithCorners.interior ↥u = Subtype.val ⁻¹' ModelWithCorners.interior M

Tactic:
ext1

Goal state after:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
u : Opens M
x✝ : ↥u
⊢ x✝ ∈ ModelWithCorners.interior ↥u ↔ x✝ ∈ Subtype.val ⁻¹' ModelWithCorners.interior M
Module: Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
n : ℕ∞ω
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝³ : TopologicalSpace G
inst✝² : ChartedSpace H G
inst✝¹ : Monoid G
inst✝ : ContMDiffMul I ∞ G
g h : G
r : 𝕜
X✝ Y✝ : LeftInvariantDerivation I G
f f' : C^∞⟮I, G; 𝓘(𝕜, 𝕜), 𝕜⟯
X Y Z : LeftInvariantDerivation I G
⊢ ⁅X + Y, Z⁆ = ⁅X, Z⁆ + ⁅Y, Z⁆

Tactic:
ext1

Goal state after:
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
n : ℕ∞ω
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝³ : TopologicalSpace G
inst✝² : ChartedSpace H G
inst✝¹ : Monoid G
inst✝ : ContMDiffMul I ∞ G
g h : G
r : 𝕜
X✝ Y✝ : LeftInvariantDerivation I G
f f' : C^∞⟮I, G; 𝓘(𝕜, 𝕜), 𝕜⟯
X Y Z : LeftInvariantDerivation I G
f✝ : C^∞⟮I, G; 𝓘(𝕜, 𝕜), 𝕜⟯
⊢ ⁅X + Y, Z⁆ f✝ = (⁅X, Z⁆ + ⁅Y, Z⁆) f✝
Module: Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
n : ℕ∞ω
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝³ : TopologicalSpace G
inst✝² : ChartedSpace H G
inst✝¹ : Monoid G
inst✝ : ContMDiffMul I ∞ G
g h : G
r : 𝕜
X✝ Y✝ : LeftInvariantDerivation I G
f f' : C^∞⟮I, G; 𝓘(𝕜, 𝕜), 𝕜⟯
X Y Z : LeftInvariantDerivation I G
⊢ ⁅X, Y + Z⁆ = ⁅X, Y⁆ + ⁅X, Z⁆

Tactic:
ext1

Goal state after:
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
n : ℕ∞ω
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝³ : TopologicalSpace G
inst✝² : ChartedSpace H G
inst✝¹ : Monoid G
inst✝ : ContMDiffMul I ∞ G
g h : G
r : 𝕜
X✝ Y✝ : LeftInvariantDerivation I G
f f' : C^∞⟮I, G; 𝓘(𝕜, 𝕜), 𝕜⟯
X Y Z : LeftInvariantDerivation I G
f✝ : C^∞⟮I, G; 𝓘(𝕜, 𝕜), 𝕜⟯
⊢ ⁅X, Y + Z⁆ f✝ = (⁅X, Y⁆ + ⁅X, Z⁆) f✝
Module: Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
n : ℕ∞ω
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝³ : TopologicalSpace G
inst✝² : ChartedSpace H G
inst✝¹ : Monoid G
inst✝ : ContMDiffMul I ∞ G
g h : G
r : 𝕜
X✝ Y : LeftInvariantDerivation I G
f f' : C^∞⟮I, G; 𝓘(𝕜, 𝕜), 𝕜⟯
X : LeftInvariantDerivation I G
⊢ ⁅X, X⁆ = 0

Tactic:
ext1

Goal state after:
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
n : ℕ∞ω
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝³ : TopologicalSpace G
inst✝² : ChartedSpace H G
inst✝¹ : Monoid G
inst✝ : ContMDiffMul I ∞ G
g h : G
r : 𝕜
X✝ Y : LeftInvariantDerivation I G
f f' : C^∞⟮I, G; 𝓘(𝕜, 𝕜), 𝕜⟯
X : LeftInvariantDerivation I G
f✝ : C^∞⟮I, G; 𝓘(𝕜, 𝕜), 𝕜⟯
⊢ ⁅X, X⁆ f✝ = 0 f✝