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

Goal state before:
pp: case neg
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)
s : AffineSubspace ℝ P
p₁ p₂ p₃ p₄ : P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
hp₃p₄ : s.SSameSide p₃ p₄
h : ¬p₁ = p₂
sp : Set (P × P × P) := (fun p ↦ (p₁, p, p₂)) '' {p | s.SSameSide p₃ p}
hc : IsConnected sp
hf : ContinuousOn (fun p ↦ ∡ p.1 p.2.1 p.2.2) sp
⊢ (∡ p₁ p₄ p₂).sign = (∡ p₁ p₃ p₂).sign

Tactic:
have hsp : ∀ p : P × P × P, p ∈ sp → ∡ p.1 p.2.1 p.2.2 ≠ 0 ∧ ∡ p.1 p.2.1 p.2.2 ≠ π := by
    intro p hp
    simp_rw [sp, Set.mem_image, Set.mem_ofPred] at hp
    obtain ⟨p', hp', rfl⟩ := hp
    dsimp only
    rw [oangle_ne_zero_and_ne_pi_iff_affineIndependent]
    exact affineIndependent_of_ne_of_mem_of_notMem_of_mem h hp₁ hp'.2.2 hp₂

Goal state after:
case neg
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)
s : AffineSubspace ℝ P
p₁ p₂ p₃ p₄ : P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
hp₃p₄ : s.SSameSide p₃ p₄
h : ¬p₁ = p₂
sp : Set (P × P × P) := (fun p ↦ (p₁, p, p₂)) '' {p | s.SSameSide p₃ p}
hc : IsConnected sp
hf : ContinuousOn (fun p ↦ ∡ p.1 p.2.1 p.2.2) sp
hsp : ∀ p ∈ sp, ∡ p.1 p.2.1 p.2.2 ≠ 0 ∧ ∡ p.1 p.2.1 p.2.2 ≠ ↑π
⊢ (∡ p₁ p₄ p₂).sign = (∡ p₁ p₃ p₂).sign
Module: Mathlib.Geometry.Euclidean.Sphere.Basic

Goal state before:
pp: P : Type u_2
inst✝ : MetricSpace P
ps : Set P
⊢ Cospherical ps ↔ ∃ s, ps ⊆ Metric.sphere s.center s.radius

Tactic:
refine ⟨fun h => ?_, fun h => ?_⟩

Goal state after:
case refine_1
P : Type u_2
inst✝ : MetricSpace P
ps : Set P
h : Cospherical ps
⊢ ∃ s, ps ⊆ Metric.sphere s.center s.radius

case refine_2
P : Type u_2
inst✝ : MetricSpace P
ps : Set P
h : ∃ s, ps ⊆ Metric.sphere s.center s.radius
⊢ Cospherical ps
Module: Mathlib.Geometry.Euclidean.Sphere.Basic

Goal state before:
pp: case inr
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
S : AffineSubspace ℝ P
inst✝¹ : Nonempty ↥S
inst✝ : S.direction.HasOrthogonalProjection
ps : Set ↥S
p₀ : ↥S
hp₀ : p₀ ∈ ps
c : P
r : ℝ
hr : ∀ p ∈ Subtype.val '' ps, dist p c = r
c' : ↥S := (orthogonalProjection S) c
p : ↥S
hp : p ∈ ps
hp_dist : dist (↑p) c = r
hp₀_dist : dist (↑p₀) c = r
hpp₀ : dist (↑p) c = dist (↑p₀) c
⊢ dist p c' = dist p₀ c'

Tactic:
exact (dist_eq_iff_dist_orthogonalProjection_eq (s := S) (p₃ := c) p.2 p₀.2).1 hpp₀

Goal state after:
Module: Mathlib.Geometry.Euclidean.Sphere.OrthRadius

Goal state before:
pp: case e'_3.e'_9
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
f : P ≃ᵃⁱ[ℝ] P
h : f s.center = s.center
e_4✝ : Real.instRing = Real.normedField.toDivisionRing.toRing
e_5✝ : inst✝³.toAddCommGroup = NormedAddCommGroup.toSeminormedAddCommGroup.toAddCommGroup
e_6✝ : inst✝².toModule ≍ inst✝².toModule
⊢ (ℝ ∙ (f p -ᵥ s.center))ᗮ = Submodule.map (↑f.toAffineEquiv).linear (ℝ ∙ (p -ᵥ s.center))ᗮ

Tactic:
convert! (Submodule.map_orthogonal_equiv (ℝ ∙ (p -ᵥ s.center)) f.linearIsometryEquiv).symm

Goal state after:
case e'_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
s : Sphere P
p : P
f : P ≃ᵃⁱ[ℝ] P
h : f s.center = s.center
e_4✝ : Real.instRing = Real.normedField.toDivisionRing.toRing
e_5✝ : inst✝³.toAddCommGroup = NormedAddCommGroup.toSeminormedAddCommGroup.toAddCommGroup
e_6✝ : inst✝².toModule ≍ inst✝².toModule
⊢ ℝ ∙ (f p -ᵥ s.center) = Submodule.map (↑f.linearIsometryEquiv.toLinearEquiv) (ℝ ∙ (p -ᵥ s.center))
Module: Mathlib.Geometry.Euclidean.Angle.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
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
t : Triangle ℝ P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
p : P
h₁ : dist p ↑((faceOpposite t i₃).orthogonalProjectionSpan p) = dist p ↑((faceOpposite t i₂).orthogonalProjectionSpan p)
h₂ : dist p ↑((faceOpposite t i₁).orthogonalProjectionSpan p) = dist p ↑((faceOpposite t i₃).orthogonalProjectionSpan p)
⊢ p ∈ affineSpan ℝ (Set.range t.points)

Tactic:
convert! AffineSubspace.mem_top ℝ V p

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
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
p : P
h₁ : dist p ↑((faceOpposite t i₃).orthogonalProjectionSpan p) = dist p ↑((faceOpposite t i₂).orthogonalProjectionSpan p)
h₂ : dist p ↑((faceOpposite t i₁).orthogonalProjectionSpan p) = dist p ↑((faceOpposite t i₃).orthogonalProjectionSpan p)
⊢ affineSpan ℝ (Set.range t.points) = ⊤
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
s : Sphere P
p q : P
h : ∠ q p s.center = π / 2
hp✝ : p ∈ s
hp_mem : p ∈ line[ℝ, p, q]
h_ortho : ⟪q -ᵥ p, p -ᵥ s.center⟫_ℝ = 0
hq : q ∈ s.orthRadius p
hp : p ∈ s.orthRadius p
⊢ p ∈ ↑(s.orthRadius p) ∧ {q} ⊆ ↑(s.orthRadius p)

Tactic:
Set.singleton_subset_iff

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
h : ∠ q p s.center = π / 2
hp✝ : p ∈ s
hp_mem : p ∈ line[ℝ, p, q]
h_ortho : ⟪q -ᵥ p, p -ᵥ s.center⟫_ℝ = 0
hq : q ∈ s.orthRadius p
hp : p ∈ s.orthRadius p
⊢ p ∈ ↑(s.orthRadius p) ∧ q ∈ ↑(s.orthRadius p)
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
⊢ s.exsphere Finset.univ = s.insphere

Tactic:
rw [← Finset.compl_empty, exsphere_compl, insphere]

Goal state after:
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
⊢ dist p₁ p₂ / (∡ p₂ p₁ s.center).cos / 2 = s.radius

Tactic:
div_right_comm,

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 : P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
h : p₁ ≠ p₂
⊢ dist p₁ p₂ / 2 / (∡ p₂ p₁ s.center).cos = s.radius
Module: Mathlib.Geometry.Euclidean.Circumcenter

Goal state before:
pp: case h.left
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
hm :
  ∀ {ι : Type u_3} [hne : Nonempty ι] [Finite ι] {p : ι → P},
    AffineIndependent ℝ p →
      ∀ (val : Fintype ι),
        Fintype.card ι = 0 →
          ∃! cs, cs.center ∈ affineSpan ℝ (Set.range p) ∧ Set.range p ⊆ Metric.sphere cs.center cs.radius
i : ι
hi : ∀ (y : ι), y = i
this : Unique ι
⊢ True ∧ {p i} ⊆ Metric.sphere (p i) 0

Tactic:
Set.singleton_subset_iff

Goal state after:
case h.left
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
hm :
  ∀ {ι : Type u_3} [hne : Nonempty ι] [Finite ι] {p : ι → P},
    AffineIndependent ℝ p →
      ∀ (val : Fintype ι),
        Fintype.card ι = 0 →
          ∃! cs, cs.center ∈ affineSpan ℝ (Set.range p) ∧ Set.range p ⊆ Metric.sphere cs.center cs.radius
i : ι
hi : ∀ (y : ι), y = i
this : Unique ι
⊢ True ∧ p i ∈ Metric.sphere (p i) 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
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
i : ι
ι2 : Type (max 0 u_3) := { x // x ≠ i }
⊢ (Finset.univ \ {x | x = i}).card = m + 1

Tactic:
Finset.filter_eq' _ i,

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
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
i : ι
ι2 : Type (max 0 u_3) := { x // x ≠ i }
⊢ (Finset.univ \ if i ∈ Finset.univ then {i} else ∅).card = m + 1
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
h : π + π ≤ angle x (x + y) + angle y (x + y)
hn : 1 ≤ n
⊢ False

Tactic:
have h' := h.trans_eq (add_comm _ _)

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
h : π + π ≤ angle x (x + y) + angle y (x + y)
hn : 1 ≤ n
h' : π + π ≤ angle y (x + y) + angle x (x + y)
⊢ False
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
⊢ ¬Collinear ℝ {t.points i₁, t.points i₂, t.points i₃}

Tactic:
←
      affineIndependent_iff_not_collinear_of_ne h₁₂ h₁₃ h₂₃

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
⊢ AffineIndependent ℝ t.points
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 : InnerProductGeometry.angle (p₁ -ᵥ p₂) (p₃ -ᵥ p₂) = InnerProductGeometry.angle (p₁ -ᵥ p₃) (p₂ -ᵥ p₃)
hpi : InnerProductGeometry.angle (p₂ -ᵥ p₁) (p₃ -ᵥ p₁) ≠ π
⊢ ‖p₁ -ᵥ p₂‖ = ‖p₁ -ᵥ p₃‖

Tactic:
← angle_neg_neg,

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
p₁ p₂ p₃ : P
h : InnerProductGeometry.angle (p₁ -ᵥ p₂) (p₃ -ᵥ p₂) = InnerProductGeometry.angle (p₁ -ᵥ p₃) (p₂ -ᵥ p₃)
hpi : InnerProductGeometry.angle (-(p₂ -ᵥ p₁)) (-(p₃ -ᵥ p₁)) ≠ π
⊢ ‖p₁ -ᵥ p₂‖ = ‖p₁ -ᵥ p₃‖
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)
p₁ p₂ p₃ p₄ : P
h : 2 • ∡ p₁ p₂ p₄ = 2 • ∡ p₁ p₃ p₄
hn : ¬Collinear ℝ {p₁, p₂, p₄}
hn' : ¬Collinear ℝ {p₁, p₃, p₄}
t₁ : Affine.Triangle ℝ P := { points := ![p₁, p₂, p₄], independent := ⋯ }
t₂ : Affine.Triangle ℝ P := { points := ![p₁, p₃, p₄], independent := ⋯ }
⊢ p₁ ∈ Metric.sphere (Affine.Simplex.circumsphere t₂).center (Affine.Simplex.circumsphere t₂).radius ∧
    p₂ ∈ Metric.sphere (Affine.Simplex.circumsphere t₂).center (Affine.Simplex.circumsphere t₂).radius ∧
      p₃ ∈ Metric.sphere (Affine.Simplex.circumsphere t₂).center (Affine.Simplex.circumsphere t₂).radius ∧
        {p₄} ⊆ Metric.sphere (Affine.Simplex.circumsphere t₂).center (Affine.Simplex.circumsphere t₂).radius

Tactic:
Set.singleton_subset_iff

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)
p₁ p₂ p₃ p₄ : P
h : 2 • ∡ p₁ p₂ p₄ = 2 • ∡ p₁ p₃ p₄
hn : ¬Collinear ℝ {p₁, p₂, p₄}
hn' : ¬Collinear ℝ {p₁, p₃, p₄}
t₁ : Affine.Triangle ℝ P := { points := ![p₁, p₂, p₄], independent := ⋯ }
t₂ : Affine.Triangle ℝ P := { points := ![p₁, p₃, p₄], independent := ⋯ }
⊢ p₁ ∈ Metric.sphere (Affine.Simplex.circumsphere t₂).center (Affine.Simplex.circumsphere t₂).radius ∧
    p₂ ∈ Metric.sphere (Affine.Simplex.circumsphere t₂).center (Affine.Simplex.circumsphere t₂).radius ∧
      p₃ ∈ Metric.sphere (Affine.Simplex.circumsphere t₂).center (Affine.Simplex.circumsphere t₂).radius ∧
        p₄ ∈ Metric.sphere (Affine.Simplex.circumsphere t₂).center (Affine.Simplex.circumsphere t₂).radius
Module: Mathlib.Geometry.Euclidean.Angle.Sphere

Goal state before:
pp: case neg
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)
p₁ p₂ p₃ p₄ : P
h : 2 • ∡ p₁ p₂ p₄ = 2 • ∡ p₁ p₃ p₄
hc : Collinear ℝ {p₁, p₂, p₄}
he : p₁ = p₄
hl : ¬Collinear ℝ {p₂, p₃, p₄}
t : Affine.Triangle ℝ P := { points := ![p₂, p₃, p₄], independent := ⋯ }
⊢ p₂ ∈ Metric.sphere (Affine.Simplex.circumsphere t).center (Affine.Simplex.circumsphere t).radius ∧
    p₃ ∈ Metric.sphere (Affine.Simplex.circumsphere t).center (Affine.Simplex.circumsphere t).radius ∧
      {p₄} ⊆ Metric.sphere (Affine.Simplex.circumsphere t).center (Affine.Simplex.circumsphere t).radius

Tactic:
Set.singleton_subset_iff

Goal state after:
case neg
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)
p₁ p₂ p₃ p₄ : P
h : 2 • ∡ p₁ p₂ p₄ = 2 • ∡ p₁ p₃ p₄
hc : Collinear ℝ {p₁, p₂, p₄}
he : p₁ = p₄
hl : ¬Collinear ℝ {p₂, p₃, p₄}
t : Affine.Triangle ℝ P := { points := ![p₂, p₃, p₄], independent := ⋯ }
⊢ p₂ ∈ Metric.sphere (Affine.Simplex.circumsphere t).center (Affine.Simplex.circumsphere t).radius ∧
    p₃ ∈ Metric.sphere (Affine.Simplex.circumsphere t).center (Affine.Simplex.circumsphere t).radius ∧
      p₄ ∈ Metric.sphere (Affine.Simplex.circumsphere t).center (Affine.Simplex.circumsphere t).radius
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.TriangleInequality

Goal state before:
pp: case pos
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y z : V
hx : ¬x = 0
hy : y = 0
⊢ angle x z ≤ angle x y + angle y z

Tactic:
simpa [hy] using angle_le_pi x z

Goal state after:
Module: Mathlib.Geometry.Euclidean.Inversion.Calculus

Goal state before:
pp: F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
c : F
R : ℝ
x : F
hx : (fun x ↦ c + x) x ≠ c
A : HasFDerivAt (fun x ↦ _root_.id x - c) (ContinuousLinearMap.id ℝ F) (c + x)
B :
  HasFDerivAt (fun y ↦ R ^ 2 * ((fun y ↦ y⁻¹) ∘ fun x ↦ ‖_root_.id x - c‖ ^ 2) y)
    (R ^ 2 •
      -((‖_root_.id (c + x) - c‖ ^ 2) ^ 2)⁻¹ • 2 • (innerSL ℝ) (_root_.id (c + x) - c) ∘SL ContinuousLinearMap.id ℝ F)
    (c + x)
⊢ HasFDerivAt (fun x ↦ (R ^ 2 * (‖x - c‖ ^ 2)⁻¹) • (x -ᵥ c) +ᵥ c) ?refine_1 (c + x)

Tactic:
exact (B.smul A).add_const c

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
⊢ Sbtw ℝ (t.points i₁) (Simplex.touchpoint t {i₂} i₂) (t.points i₃)

Tactic:
by
  rw [← t.mem_interior_face_iff_sbtw h₁₃]
  convert! t.touchpoint_singleton_mem_interior_faceOpposite i₂
  rw [Affine.Simplex.faceOpposite]
  convert! rfl using 2
  decide +revert

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
m n : ℕ
s : Simplex ℝ P n
e : Fin (n + 1) ≃ Fin (m + 1)
h : n = m
i : Fin (m + 1)
⊢ (s.reindex e).eulerPoint i = (s.eulerPoint ∘ ⇑e.symm) i

Tactic:
simp [eulerPoint, h]

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
m n : ℕ
s : Simplex ℝ P n
e : Fin (n + 1) ≃ Fin (m + 1)
⊢ (↑(m + 1) / ↑(m - 1)) • (Finset.centroid ℝ univ (s.reindex e).points -ᵥ (s.reindex e).circumcenter) +ᵥ
      (s.reindex e).circumcenter =
    (↑(n + 1) / ↑(n - 1)) • (Finset.centroid ℝ univ s.points -ᵥ s.circumcenter) +ᵥ s.circumcenter

Tactic:
circumcenter_reindex,

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
m n : ℕ
s : Simplex ℝ P n
e : Fin (n + 1) ≃ Fin (m + 1)
⊢ (↑(m + 1) / ↑(m - 1)) • (Finset.centroid ℝ univ (s.reindex e).points -ᵥ s.circumcenter) +ᵥ s.circumcenter =
    (↑(n + 1) / ↑(n - 1)) • (Finset.centroid ℝ univ s.points -ᵥ s.circumcenter) +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)
e : Fin (n + 3) ≃ Fin (n + 3)
i₁ i₂ : Fin (n + 3)
⊢ (s.reindex e).mongePlane i₁ i₂ = s.mongePlane (e.symm i₁) (e.symm i₂)

Tactic:
mongePlane,

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)
e : Fin (n + 3) ≃ Fin (n + 3)
i₁ i₂ : Fin (n + 3)
⊢ mk' (Finset.centroid ℝ {i₁, i₂}ᶜ (s.reindex e).points) (ℝ ∙ ((s.reindex e).points i₁ -ᵥ (s.reindex e).points i₂))ᗮ ⊓
      affineSpan ℝ (Set.range (s.reindex e).points) =
    mk' (Finset.centroid ℝ {e.symm i₁, e.symm i₂}ᶜ s.points) (ℝ ∙ (s.points (e.symm i₁) -ᵥ s.points (e.symm i₂)))ᗮ ⊓
      affineSpan ℝ (Set.range s.points)
Module: Mathlib.Geometry.Euclidean.Sphere.SecondInter

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 : P
hp : p ∈ s
v : V
p' : P
hp' : p' ∈ AffineSubspace.mk' p (ℝ ∙ v)
⊢ p' = p ∨ p' = s.secondInter p v ↔ p' ∈ s

Tactic:
refine ⟨fun h => ?_, fun h => ?_⟩

Goal state after:
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
s : Sphere P
p : P
hp : p ∈ s
v : V
p' : P
hp' : p' ∈ AffineSubspace.mk' p (ℝ ∙ v)
h : p' = p ∨ p' = s.secondInter p v
⊢ p' ∈ s

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
s : Sphere P
p : P
hp : p ∈ s
v : V
p' : P
hp' : p' ∈ AffineSubspace.mk' p (ℝ ∙ v)
h : p' ∈ s
⊢ p' = p ∨ p' = s.secondInter p v
Module: Mathlib.Geometry.Euclidean.Sphere.SecondInter

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 : Sphere P
n : ℕ
inst✝ : n.AtLeastTwo
sx : Affine.Simplex ℝ P n
i : Fin (n + 1)
hi : sx.points i ∈ s
hsx : ∀ (j : Fin (n + 1)), dist (sx.points j) s.center ≤ s.radius
w : Fin (n + 1) → ℝ
hw : ∑ i, w i = 1
hw01 : ∀ (i : Fin (n + 1)), w i ∈ Set.Ioo 0 1
r : ℝ := (1 - w i)⁻¹
hrpos : 0 < r
p' : P := (AffineMap.lineMap (sx.points i) ((Finset.affineCombination ℝ Finset.univ sx.points) w)) r
hp' : p' -ᵥ sx.points i = r • ((Finset.affineCombination ℝ Finset.univ sx.points) w -ᵥ sx.points i)
⊢ p' ∈ (sx.faceOpposite i).interior

Tactic:
p',

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 : Sphere P
n : ℕ
inst✝ : n.AtLeastTwo
sx : Affine.Simplex ℝ P n
i : Fin (n + 1)
hi : sx.points i ∈ s
hsx : ∀ (j : Fin (n + 1)), dist (sx.points j) s.center ≤ s.radius
w : Fin (n + 1) → ℝ
hw : ∑ i, w i = 1
hw01 : ∀ (i : Fin (n + 1)), w i ∈ Set.Ioo 0 1
r : ℝ := (1 - w i)⁻¹
hrpos : 0 < r
p' : P := (AffineMap.lineMap (sx.points i) ((Finset.affineCombination ℝ Finset.univ sx.points) w)) r
hp' : p' -ᵥ sx.points i = r • ((Finset.affineCombination ℝ Finset.univ sx.points) w -ᵥ sx.points i)
⊢ (AffineMap.lineMap (sx.points i) ((Finset.affineCombination ℝ Finset.univ sx.points) w)) r ∈
    (sx.faceOpposite i).interior
Module: Mathlib.Geometry.Euclidean.Similarity

Goal state before:
pp: case pos.inr
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')
h2 : c' = b'
h_dist_cb : c = b
⊢ ∠ a b b = ∠ a' b' b'

Tactic:
angle_self_right

Goal state after:
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
s : Sphere P
p a b : P
hr : 0 ≤ s.radius
hp : p ∈ line[ℝ, a, b]
ha : a ∈ s
hb : b ∈ s
hle : dist p s.center ≤ s.radius
⊢ |s.power p| = -s.power p

Tactic:
abs_of_nonpos <| (power_nonpos_iff_dist_center_le_radius hr).mpr hle

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
s : Sphere P
p a b : P
hr : 0 ≤ s.radius
hp : p ∈ line[ℝ, a, b]
ha : a ∈ s
hb : b ∈ s
hle : dist p s.center ≤ s.radius
⊢ -s.power p = -s.power p
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
⊢ dist (circumcenter t) ((reflection (affineSpan ℝ (t.points '' {i₁, i₂}))) t.orthocenter) = circumradius t

Tactic:
by
  rw [EuclideanGeometry.dist_reflection, dist_comm, dist_orthocenter_reflection_circumcenter t h]

Goal state after: