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
⊢ p -ᵥ p₁ ∈ line[ℝ, p₁, p₃].directionᗮ

Tactic:
rw [Submodule.mem_orthogonal']

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 ∈ line[ℝ, p₁, p₃].direction, inner ℝ (p -ᵥ p₁) u = 0
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
r : ℝ
⊢ inner ℝ (p -ᵥ p₁) (r • (p₁ -ᵥ p₃)) = 0

Tactic:
rw [inner_smul_right, ← inner_neg_neg, inner_neg_left]

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
r : ℝ
⊢ r * -inner ℝ (p -ᵥ p₁) (-(p₁ -ᵥ p₃)) = 0
Module: Mathlib.Geometry.Euclidean.Sphere.Basic

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 p : P
s : Sphere P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
o : P := s.center
t : ℝ
hpt : (AffineMap.lineMap p₁ p₂) t = p
ht₀ : 0 ≤ t
ht₁ : t ≤ 1
hne₁ : p ≠ p₁
hne₂ : p ≠ p₂
ht₀' : 0 < t
ht₁' : t < 1
u : V := p₁ -ᵥ o
v : V := p₂ -ᵥ o
hu : ‖u‖ = s.radius
hv : ‖v‖ = s.radius
huv : u ≠ v
⊢ p -ᵥ o = (1 - t) • u + t • v

Tactic:
show p = (AffineMap.lineMap p₁ p₂) t from hpt.symm,

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 p : P
s : Sphere P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
o : P := s.center
t : ℝ
hpt : (AffineMap.lineMap p₁ p₂) t = p
ht₀ : 0 ≤ t
ht₁ : t ≤ 1
hne₁ : p ≠ p₁
hne₂ : p ≠ p₂
ht₀' : 0 < t
ht₁' : t < 1
u : V := p₁ -ᵥ o
v : V := p₂ -ᵥ o
hu : ‖u‖ = s.radius
hv : ‖v‖ = s.radius
huv : u ≠ v
⊢ (AffineMap.lineMap p₁ p₂) t -ᵥ o = (1 - t) • u + t • v
Module: Mathlib.Geometry.Euclidean.Sphere.Tangent

Goal state before:
pp: case refine_2.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
as : AffineSubspace ℝ P
hr : s.radius = 0
hm : s.center ∈ as
⊢ s.radius = 0

Tactic:
hr

Goal state after:
case refine_2.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
as : AffineSubspace ℝ P
hr : s.radius = 0
hm : s.center ∈ as
⊢ 0 = 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
as : AffineSubspace ℝ P
p : P
hpar : as ∥ s.orthRadius p
hp : p ∈ s
q : P
hqs : q ∈ s
hqas : q ∈ as
hqo : as ≤ s.orthRadius q
⊢ as = s.orthRadius p ∨ as = s.orthRadius ((Equiv.pointReflection s.center) p)

Tactic:
have hd := direction_le hqo

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
as : AffineSubspace ℝ P
p : P
hpar : as ∥ s.orthRadius p
hp : p ∈ s
q : P
hqs : q ∈ s
hqas : q ∈ as
hqo : as ≤ s.orthRadius q
hd : as.direction ≤ (s.orthRadius q).direction
⊢ as = s.orthRadius p ∨ as = s.orthRadius ((Equiv.pointReflection s.center) p)
Module: Mathlib.Geometry.Euclidean.Angle.Bisector

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
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p p₁ p₂ p₃ : P
ha : AffineIndependent ℝ ![p₁, p₂, p₃]
h : dist p ↑((orthogonalProjection line[ℝ, p₁, p₂]) p) = dist p ↑((orthogonalProjection line[ℝ, p₁, p₃]) p)
ho : ↑((orthogonalProjection line[ℝ, p₁, p₂]) p) = ↑((orthogonalProjection line[ℝ, p₁, p₃]) p)
hs : ↑((orthogonalProjection (line[ℝ, p₁, p₂] ⊔ line[ℝ, p₁, p₃])) p) = ↑((orthogonalProjection line[ℝ, p₁, p₂]) p)
hinf : line[ℝ, p₁, p₂] ⊓ line[ℝ, p₁, p₃] = affineSpan ℝ {p₁}
hsup : line[ℝ, p₁, p₂] ⊔ line[ℝ, p₁, p₃] = ⊤
hp : ↑((orthogonalProjection (line[ℝ, p₁, p₂] ⊔ line[ℝ, p₁, p₃])) p) = p₁
⊢ ↑((orthogonalProjection (line[ℝ, p₁, p₂] ⊔ line[ℝ, p₁, p₃])) p) = p

Tactic:
orthogonalProjection_eq_self_iff,

Goal state after:
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
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p p₁ p₂ p₃ : P
ha : AffineIndependent ℝ ![p₁, p₂, p₃]
h : dist p ↑((orthogonalProjection line[ℝ, p₁, p₂]) p) = dist p ↑((orthogonalProjection line[ℝ, p₁, p₃]) p)
ho : ↑((orthogonalProjection line[ℝ, p₁, p₂]) p) = ↑((orthogonalProjection line[ℝ, p₁, p₃]) p)
hs : ↑((orthogonalProjection (line[ℝ, p₁, p₂] ⊔ line[ℝ, p₁, p₃])) p) = ↑((orthogonalProjection line[ℝ, p₁, p₂]) p)
hinf : line[ℝ, p₁, p₂] ⊓ line[ℝ, p₁, p₃] = affineSpan ℝ {p₁}
hsup : line[ℝ, p₁, p₂] ⊔ line[ℝ, p₁, p₃] = ⊤
hp : ↑((orthogonalProjection (line[ℝ, p₁, p₂] ⊔ line[ℝ, p₁, p₃])) p) = p₁
⊢ p ∈ line[ℝ, p₁, p₂] ⊔ line[ℝ, p₁, p₃]
Module: Mathlib.Geometry.Euclidean.Sphere.OrthRadius

Goal state before:
pp: case mp
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 : Metric.sphere s.center s.radius ∩ ↑(s.orthRadius p) = {q}
hq : q ∈ Metric.sphere s.center s.radius ∩ ↑(s.orthRadius p)
⊢ q = p ∧ dist p s.center = s.radius

Tactic:
have h' (q' : P) : q' ∈ (s ∩ s.orthRadius p : Set P) ↔ q' = q := by simp [h]

Goal state after:
case mp
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 : Metric.sphere s.center s.radius ∩ ↑(s.orthRadius p) = {q}
hq : q ∈ Metric.sphere s.center s.radius ∩ ↑(s.orthRadius p)
h' : ∀ (q' : P), q' ∈ Metric.sphere s.center s.radius ∩ ↑(s.orthRadius p) ↔ q' = q
⊢ q = p ∧ dist p s.center = s.radius
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 : P
hp : p ∈ s
q : P
hqs : q ∈ s
r : ℝ
hr : q -ᵥ s.center = r • (p -ᵥ s.center)
hrad : s.radius ≠ 0
hpar : s.orthRadius q ∥ s.orthRadius p
hqas : q ∈ s.orthRadius q
hqo : s.orthRadius q ≤ s.orthRadius q
hr0 : r ≠ 0
⊢ ‖q -ᵥ s.center‖ = ‖r • (p -ᵥ s.center)‖

Tactic:
hr

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 : P
hp : p ∈ s
q : P
hqs : q ∈ s
r : ℝ
hr : q -ᵥ s.center = r • (p -ᵥ s.center)
hrad : s.radius ≠ 0
hpar : s.orthRadius q ∥ s.orthRadius p
hqas : q ∈ s.orthRadius q
hqo : s.orthRadius q ≤ s.orthRadius q
hr0 : r ≠ 0
⊢ ‖r • (p -ᵥ s.center)‖ = ‖r • (p -ᵥ s.center)‖
Module: Mathlib.Geometry.Euclidean.Angle.Bisector

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 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p p₁ p₂ p₃ : P
ha : AffineIndependent ℝ ![p₁, p₂, p₃]
h : dist p ↑((orthogonalProjection line[ℝ, p₁, p₂]) p) = dist p ↑((orthogonalProjection line[ℝ, p₁, p₃]) p)
⊢ 2 • ∡ p₂ p₁ p = 2 • ∡ p p₁ p₃

Tactic:
by_cases ho : (orthogonalProjection line[ℝ, p₁, p₂] p : P) =
      orthogonalProjection line[ℝ, p₁, p₃] p
  · suffices p = p₁ by simp [this]
    have hs := orthogonalProjection_sup_of_orthogonalProjection_eq ho
    have hinf : line[ℝ, p₁, p₂] ⊓ line[ℝ, p₁, p₃] = affineSpan ℝ {p₁} := by
      convert! (ha.inf_affineSpan_eq_affineSpan_inter {0, 1} {0, 2})
      · simp [Set.image_insert_eq]
      · simp [Set.image_insert_eq]
      · suffices {p₁} = ![p₁, p₂, p₃] '' {0} by grind
        simp
    have hsup : line[ℝ, p₁, p₂] ⊔ line[ℝ, p₁, p₃] = ⊤ := by
      rw [← AffineSubspace.span_union]
      convert! ha.affineSpan_eq_top_iff_card_eq_finrank_add_one.2 ?_
      · simp
        grind
      · simpa using Fact.out
    have hp : orthogonalProjection (line[ℝ, p₁, p₂]) p = p₁ := by
      suffices (orthogonalProjection (line[ℝ, p₁, p₂]) p : P) ∈ affineSpan ℝ {p₁} by
        simpa using this
      have hi : (orthogonalProjection (line[ℝ, p₁, p₂]) p : P) ∈
          line[ℝ, p₁, p₂] ⊓ line[ℝ, p₁, p₃] :=
        ⟨orthogonalProjection_mem _, ho ▸ orthogonalProjection_mem _⟩
      rwa [hinf] at hi
    rw [← orthogonalProjection_sup_of_orthogonalProjection_eq ho] at hp
    rw [← hp, eq_comm, orthogonalProjection_eq_self_iff, hsup]
    exact AffineSubspace.mem_top ℝ V p
  · have hp := oangle_eq_of_dist_orthogonalProjection_eq
      (left_mem_affineSpan_pair _ _ _) (left_mem_affineSpan_pair _ _ _) ho h
    have h₂₁ : p₂ ≠ p₁ := ha.injective.ne (by decide : (1 : Fin 3) ≠ 0)
    have h₃₁ : p₃ ≠ p₁ := ha.injective.ne (by decide : (2 : Fin 3) ≠ 0)
    have hp₁ : orthogonalProjection line[ℝ, p₁, p₂] p ≠ p₁ := by
      intro hp
      rw [hp, eq_comm, dist_orthogonalProjection_eq_dist_iff_eq_of_mem
        (left_mem_affineSpan_pair ℝ _ p₃)] at h
      grind
    have hp₂ : orthogonalProjection line[ℝ, p₁, p₃] p ≠ p₁ := by
      intro hp
      rw [hp, dist_orthogonalProjection_eq_dist_iff_eq_of_mem
          (left_mem_affineSpan_pair ℝ _ p₂)] at h
      grind
    rw [← (collinear_insert_of_mem_affineSpan_pair
             (orthogonalProjection_mem p)).two_zsmul_oangle_eq_left hp₁ h₂₁,
        ← (collinear_insert_of_mem_affineSpan_pair
             (orthogonalProjection_mem p)).two_zsmul_oangle_eq_right hp₂ h₃₁, hp]

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

Goal state before:
pp: case e'_3
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
q : P
hqs : q ∈ s
hrad : s.radius ≠ 0
hpar : s.orthRadius q ∥ s.orthRadius p
hqas : q ∈ s.orthRadius q
hqo : s.orthRadius q ≤ s.orthRadius q
hr : q = -1 • (p -ᵥ s.center) +ᵥ s.center
hr0 : -1 ≠ 0
hr' : |(-1)| = 1
⊢ (Equiv.pointReflection s.center) p = q

Tactic:
hr

Goal state after:
case e'_3
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
q : P
hqs : q ∈ s
hrad : s.radius ≠ 0
hpar : s.orthRadius q ∥ s.orthRadius p
hqas : q ∈ s.orthRadius q
hqo : s.orthRadius q ≤ s.orthRadius q
hr : q = -1 • (p -ᵥ s.center) +ᵥ s.center
hr0 : -1 ≠ 0
hr' : |(-1)| = 1
⊢ (Equiv.pointReflection s.center) p = -1 • (p -ᵥ s.center) +ᵥ s.center
Module: Mathlib.Geometry.Euclidean.Sphere.Tangent

Goal state before:
pp: case refine_2.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
s₁ s₂ : Sphere P
hr : s₁.radius = 0
hc : s₁.center ∈ s₂
⊢ s₁.radius = 0

Tactic:
hr

Goal state after:
case refine_2.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
s₁ s₂ : Sphere P
hr : s₁.radius = 0
hc : s₁.center ∈ s₂
⊢ 0 = 0
Module: Mathlib.Geometry.Euclidean.Sphere.Tangent

Goal state before:
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
s₁ s₂ : Sphere P
⊢ dist s₁.center s₂.center = s₁.radius + s₂.radius ∧ 0 ≤ s₁.radius ∧ 0 ≤ s₂.radius → s₁.IsExtTangent s₂

Tactic:
rintro ⟨h, h₁, h₂⟩

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
s₁ s₂ : Sphere P
h : dist s₁.center s₂.center = s₁.radius + s₂.radius
h₁ : 0 ≤ s₁.radius
h₂ : 0 ≤ s₂.radius
⊢ s₁.IsExtTangent s₂
Module: Mathlib.Geometry.Euclidean.Sphere.Tangent

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
inst✝ : Nontrivial V
s₁ s₂ : Sphere P
⊢ dist s₁.center s₂.center = s₂.radius - s₁.radius ∧ 0 ≤ s₁.radius ∧ 0 ≤ s₂.radius → s₁.IsIntTangent s₂

Tactic:
rintro ⟨h, h₁, h₂⟩

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
inst✝ : Nontrivial V
s₁ s₂ : Sphere P
h : dist s₁.center s₂.center = s₂.radius - s₁.radius
h₁ : 0 ≤ s₁.radius
h₂ : 0 ≤ s₂.radius
⊢ s₁.IsIntTangent s₂
Module: Mathlib.Geometry.Euclidean.Angle.Incenter

Goal state before:
pp: case inr
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
signs : Finset (Fin 3)
h₁ : ∡ (t.points i₂) (t.points i₁) (excenter t signs) = ∡ (excenter t signs) (t.points i₁) (t.points i₃)
h₂ : ∡ (t.points i₃) (t.points i₂) (excenter t signs) = ∡ (excenter t signs) (t.points i₂) (t.points i₁)
h₂' : excenter t signs = incenter t ∨ excenter t signs = excenter t {i₂}
h₁' : excenter t signs = excenter t {i₁}
⊢ excenter t signs = incenter t

Tactic:
rcases h₂' with h₂' | h₂'

Goal state after:
case inr.inl
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
signs : Finset (Fin 3)
h₁ : ∡ (t.points i₂) (t.points i₁) (excenter t signs) = ∡ (excenter t signs) (t.points i₁) (t.points i₃)
h₂ : ∡ (t.points i₃) (t.points i₂) (excenter t signs) = ∡ (excenter t signs) (t.points i₂) (t.points i₁)
h₁' : excenter t signs = excenter t {i₁}
h₂' : excenter t signs = incenter t
⊢ excenter t signs = incenter t

case inr.inr
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
signs : Finset (Fin 3)
h₁ : ∡ (t.points i₂) (t.points i₁) (excenter t signs) = ∡ (excenter t signs) (t.points i₁) (t.points i₃)
h₂ : ∡ (t.points i₃) (t.points i₂) (excenter t signs) = ∡ (excenter t signs) (t.points i₂) (t.points i₁)
h₁' : excenter t signs = excenter t {i₁}
h₂' : excenter t signs = excenter t {i₂}
⊢ excenter t signs = incenter t
Module: Mathlib.Geometry.Euclidean.Angle.Incenter

Goal state before:
pp: case inl
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
signs : Finset (Fin 3)
h₁ : ∡ (t.points i₂) (t.points i₁) (excenter t signs) = ∡ (excenter t signs) (t.points i₁) (t.points i₃)
h₂ : ∡ (t.points i₃) (t.points i₂) (excenter t signs) = ∡ (excenter t signs) (t.points i₂) (t.points i₁) + ↑π
h₂' : excenter t signs = excenter t {i₃} ∨ excenter t signs = excenter t {i₁}
h₁' : excenter t signs = incenter t
⊢ excenter t signs = excenter t {i₁}

Tactic:
rcases h₂' with h₂' | h₂'

Goal state after:
case inl.inl
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
signs : Finset (Fin 3)
h₁ : ∡ (t.points i₂) (t.points i₁) (excenter t signs) = ∡ (excenter t signs) (t.points i₁) (t.points i₃)
h₂ : ∡ (t.points i₃) (t.points i₂) (excenter t signs) = ∡ (excenter t signs) (t.points i₂) (t.points i₁) + ↑π
h₁' : excenter t signs = incenter t
h₂' : excenter t signs = excenter t {i₃}
⊢ excenter t signs = excenter t {i₁}

case inl.inr
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
signs : Finset (Fin 3)
h₁ : ∡ (t.points i₂) (t.points i₁) (excenter t signs) = ∡ (excenter t signs) (t.points i₁) (t.points i₃)
h₂ : ∡ (t.points i₃) (t.points i₂) (excenter t signs) = ∡ (excenter t signs) (t.points i₂) (t.points i₁) + ↑π
h₁' : excenter t signs = incenter t
h₂' : excenter t signs = excenter t {i₁}
⊢ excenter t signs = excenter t {i₁}
Module: Mathlib.Geometry.Euclidean.Circumcenter

Goal state before:
pp: case h.left.right.inl
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
hnps : ps.Nonempty
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)
p₁ : P
hp₁ : p₁ = p
⊢ dist cc ↑((orthogonalProjection s) p) * dist cc ↑((orthogonalProjection s) p) +
      (1 - ycc₂ / y) * (1 - ycc₂ / y) * (dist p ↑((orthogonalProjection s) p) * dist p ↑((orthogonalProjection s) p)) =
    cr * cr + ycc₂ * ycc₂

Tactic:
simp only [ycc₂]

Goal state after:
case h.left.right.inl
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
hnps : ps.Nonempty
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)
p₁ : P
hp₁ : p₁ = p
⊢ dist cc ↑((orthogonalProjection s) p) * dist cc ↑((orthogonalProjection s) p) +
      (1 - (x * x + y * y - cr * cr) / (2 * y) / y) * (1 - (x * x + y * y - cr * cr) / (2 * y) / y) *
        (dist p ↑((orthogonalProjection s) p) * dist p ↑((orthogonalProjection s) p)) =
    cr * cr + (x * x + y * y - cr * cr) / (2 * y) * ((x * x + y * y - cr * cr) / (2 * y))
Module: Mathlib.Geometry.Euclidean.Angle.Incenter

Goal state before:
pp: case inl
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
signs : Finset (Fin 3)
h₁ : ∡ (t.points i₂) (t.points i₁) (excenter t signs) = ∡ (excenter t signs) (t.points i₁) (t.points i₃) + ↑π
h₂ : ∡ (t.points i₃) (t.points i₂) (excenter t signs) = ∡ (excenter t signs) (t.points i₂) (t.points i₁) + ↑π
h₂' : excenter t signs = excenter t {i₃} ∨ excenter t signs = excenter t {i₁}
h₁' : excenter t signs = excenter t {i₂}
⊢ excenter t signs = excenter t {i₃}

Tactic:
rcases h₂' with h₂' | h₂'

Goal state after:
case inl.inl
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
signs : Finset (Fin 3)
h₁ : ∡ (t.points i₂) (t.points i₁) (excenter t signs) = ∡ (excenter t signs) (t.points i₁) (t.points i₃) + ↑π
h₂ : ∡ (t.points i₃) (t.points i₂) (excenter t signs) = ∡ (excenter t signs) (t.points i₂) (t.points i₁) + ↑π
h₁' : excenter t signs = excenter t {i₂}
h₂' : excenter t signs = excenter t {i₃}
⊢ excenter t signs = excenter t {i₃}

case inl.inr
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
signs : Finset (Fin 3)
h₁ : ∡ (t.points i₂) (t.points i₁) (excenter t signs) = ∡ (excenter t signs) (t.points i₁) (t.points i₃) + ↑π
h₂ : ∡ (t.points i₃) (t.points i₂) (excenter t signs) = ∡ (excenter t signs) (t.points i₂) (t.points i₁) + ↑π
h₁' : excenter t signs = excenter t {i₂}
h₂' : excenter t signs = excenter t {i₁}
⊢ excenter t signs = excenter t {i₃}
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)
⊢ SignType.sign (s.excenterWeights {i} i) = -1

Tactic:
by
  simp_rw [excenterWeights, Pi.smul_apply, smul_eq_mul, sign_mul]
  convert! one_mul _
  · rw [sign_eq_one_iff, inv_pos]
    exact s.sum_excenterWeightsUnnorm_singleton_pos i
  · simp [excenterWeightsUnnorm]

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

Goal state before:
pp: case inr.inr.inr
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hy : y ≠ 0
hx : x ≠ 0
hxy : x ≠ y
h_sin : ∀ (x y : V), x ≠ 0 → y ≠ 0 → Real.sin (angle x y) = √(⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫) / (‖x‖ * ‖y‖)
hsub : x - y ≠ 0
⊢ √(⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫) / (‖x‖ * ‖y‖) * ‖x‖ =
    √(⟪y, y⟫ * (⟪x, x⟫ - ⟪x, y⟫ - (⟪x, y⟫ - ⟪y, y⟫)) - (⟪x, y⟫ - ⟪y, y⟫) * (⟪x, y⟫ - ⟪y, y⟫)) / (‖y‖ * ‖x - y‖) *
      ‖x - y‖

Tactic:
field_simp

Goal state after:
case inr.inr.inr
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hy : y ≠ 0
hx : x ≠ 0
hxy : x ≠ y
h_sin : ∀ (x y : V), x ≠ 0 → y ≠ 0 → Real.sin (angle x y) = √(⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫) / (‖x‖ * ‖y‖)
hsub : x - y ≠ 0
⊢ √(⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ ^ 2) = √(⟪y, y⟫ * (⟪x, x⟫ - ⟪x, y⟫ - (⟪x, y⟫ - ⟪y, y⟫)) - (⟪x, y⟫ - ⟪y, y⟫) ^ 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
signs : Finset (Fin (n + 1))
⊢ (s.exsphere signsᶜ).center = (s.exsphere signs).center

Tactic:
exsphere_compl

Goal state after: