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

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
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ p₄ : P
hp₁ : p₁ ≠ p₂
hp₄ : p₄ ≠ p₂
h : Wbtw ℝ p₂ p₁ p₄ ∨ Wbtw ℝ p₂ p₄ p₁
⊢ SameRay ℝ (p₁ -ᵥ p₂) (p₄ -ᵥ p₂)

Tactic:
rcases h with h | h
    · exact h.sameRay_vsub_left
    · exact h.sameRay_vsub_left.symm

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p₁ p₂ : P
h : s.IsDiameter p₁ p₂
⊢ p₁ = p₂ ↔ s.radius = 0

Tactic:
rw [← dist_eq_zero, h.dist_left_right]

Goal state after:
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p₁ p₂ : P
h : s.IsDiameter p₁ p₂
⊢ 2 * s.radius = 0 ↔ s.radius = 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
s : Set P
hs : Cospherical s
p : Fin 3 → P
hps : Set.range p ⊆ s
hpi : Function.Injective p
hc : Collinear ℝ (Set.range p)
⊢ False

Tactic:
collinear_iff_of_mem (Set.mem_range_self (0 : Fin 3))

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 : Set P
hs : Cospherical s
p : Fin 3 → P
hps : Set.range p ⊆ s
hpi : Function.Injective p
hc : ∃ v, ∀ p_1 ∈ Set.range p, ∃ r, p_1 = r • v +ᵥ p 0
⊢ False
Module: Mathlib.Geometry.Euclidean.Basic

Goal state before:
pp: case refine_1
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✝ : FiniteDimensional ℝ ↥s.direction
hd : finrank ℝ ↥s.direction = 2
c₁ c₂ p₁ p₂ p : P
hc₁s : c₁ ∈ s
hc₂s : c₂ ∈ s
hp₁s : p₁ ∈ s
hp₂s : p₂ ∈ s
hps : p ∈ s
r₁ r₂ : ℝ
hc : c₁ ≠ c₂
hp : p₁ ≠ p₂
hp₁c₁ : dist p₁ c₁ = r₁
hp₂c₁ : dist p₂ c₁ = r₁
hpc₁ : dist p c₁ = r₁
hp₁c₂ : dist p₁ c₂ = r₂
hp₂c₂ : dist p₂ c₂ = r₂
hpc₂ : dist p c₂ = r₂
ho : ⟪c₂ -ᵥ c₁, p₂ -ᵥ p₁⟫ = 0
hop : ⟪c₂ -ᵥ c₁, p -ᵥ p₁⟫ = 0
b : Fin 2 → V := ![c₂ -ᵥ c₁, p₂ -ᵥ p₁]
hb : LinearIndependent ℝ b
⊢ Submodule.span ℝ (Set.range b) ≤ s.direction

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
s : AffineSubspace ℝ P
inst✝ : FiniteDimensional ℝ ↥s.direction
hd : finrank ℝ ↥s.direction = 2
c₁ c₂ p₁ p₂ p : P
hc₁s : c₁ ∈ s
hc₂s : c₂ ∈ s
hp₁s : p₁ ∈ s
hp₂s : p₂ ∈ s
hps : p ∈ s
r₁ r₂ : ℝ
hc : c₁ ≠ c₂
hp : p₁ ≠ p₂
hp₁c₁ : dist p₁ c₁ = r₁
hp₂c₁ : dist p₂ c₁ = r₁
hpc₁ : dist p c₁ = r₁
hp₁c₂ : dist p₁ c₂ = r₂
hp₂c₂ : dist p₂ c₂ = r₂
hpc₂ : dist p c₂ = r₂
ho : ⟪c₂ -ᵥ c₁, p₂ -ᵥ p₁⟫ = 0
hop : ⟪c₂ -ᵥ c₁, p -ᵥ p₁⟫ = 0
b : Fin 2 → V := ![c₂ -ᵥ c₁, p₂ -ᵥ p₁]
hb : LinearIndependent ℝ b
⊢ finrank ℝ ↥(Submodule.span ℝ (Set.range b)) = finrank ℝ ↥s.direction

Tactic:
· rw [Submodule.span_le, Set.range_subset_iff]
      intro i
      fin_cases i
      · exact vsub_mem_direction hc₂s hc₁s
      · exact vsub_mem_direction hp₂s hp₁s

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
s : AffineSubspace ℝ P
inst✝ : FiniteDimensional ℝ ↥s.direction
hd : finrank ℝ ↥s.direction = 2
c₁ c₂ p₁ p₂ p : P
hc₁s : c₁ ∈ s
hc₂s : c₂ ∈ s
hp₁s : p₁ ∈ s
hp₂s : p₂ ∈ s
hps : p ∈ s
r₁ r₂ : ℝ
hc : c₁ ≠ c₂
hp : p₁ ≠ p₂
hp₁c₁ : dist p₁ c₁ = r₁
hp₂c₁ : dist p₂ c₁ = r₁
hpc₁ : dist p c₁ = r₁
hp₁c₂ : dist p₁ c₂ = r₂
hp₂c₂ : dist p₂ c₂ = r₂
hpc₂ : dist p c₂ = r₂
ho : ⟪c₂ -ᵥ c₁, p₂ -ᵥ p₁⟫ = 0
hop : ⟪c₂ -ᵥ c₁, p -ᵥ p₁⟫ = 0
b : Fin 2 → V := ![c₂ -ᵥ c₁, p₂ -ᵥ p₁]
hb : LinearIndependent ℝ b
⊢ finrank ℝ ↥(Submodule.span ℝ (Set.range b)) = finrank ℝ ↥s.direction
Module: Mathlib.Geometry.Euclidean.Sphere.Basic

Goal state before:
pp: case pos
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p₁ p₂ : P
hp₁ : dist p₂ s.center = s.radius
hp₂ : dist p₂ s.center < s.radius
h : p₁ = p₂
⊢ 0 < ⟪p₁ -ᵥ p₂, p₁ -ᵥ s.center⟫

Tactic:
exact False.elim (hp₂.ne hp₁)

Goal state after:
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
s : Sphere P
p₁ p₂ : P
⊢ s.IsDiameter p₁ p₂ ↔ p₁ ∈ s ∧ p₂ ∈ s ∧ dist p₁ p₂ = 2 * s.radius

Tactic:
refine ⟨fun h ↦ ⟨h.left_mem, h.right_mem, h.dist_left_right⟩, fun ⟨h₁, h₂, hr⟩ ↦ ⟨h₁, ?_⟩⟩
  rw [midpoint_eq_iff, AffineEquiv.pointReflection_apply, eq_comm, eq_vadd_iff_vsub_eq]
  apply eq_of_norm_eq_of_norm_add_eq
  · simp_rw [← dist_eq_norm_vsub, mem_sphere'.1 h₁, mem_sphere.1 h₂]
  · simp_rw [vsub_add_vsub_cancel, ← dist_eq_norm_vsub, mem_sphere'.1 h₁, mem_sphere.1 h₂]
    rw [dist_comm, hr, two_mul]

Goal state after:
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
s₁ s₂ : Sphere P
⊢ s₁.commonIntTangents s₂ ∪ s₁.commonExtTangents s₂ = s₁.commonTangents s₂

Tactic:
ext as
  rw [Set.mem_union, mem_commonIntTangents_iff, mem_commonExtTangents_iff, ← and_or_left,
    and_iff_left_iff_imp]
  rintro -
  by_cases! h : ∃ p ∈ as, Wbtw ℝ s₁.center p s₂.center
  · exact .inl h
  · refine .inr ?_
    rintro p hp
    exact mt Sbtw.wbtw (h p hp)

Goal state after:
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
s₁ s₂ : Sphere P
h : s₁.IsExtTangent s₂
⊢ s₂.IsExtTangent s₁

Tactic:
rcases h with ⟨p, hp⟩

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
s₁ s₂ : Sphere P
p : P
hp : s₁.IsExtTangentAt s₂ p
⊢ s₂.IsExtTangent s₁
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
s₁ s₂ : Sphere P
p : P
hp : s₁.IsExtTangentAt s₂ p
⊢ s₂.IsExtTangent s₁

Tactic:
exact ⟨p, hp.symm⟩

Goal state after:
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
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 = incenter t ∨ excenter t signs = excenter t {i₁}
h₂' : excenter t signs = incenter t ∨ excenter t signs = excenter t {i₂}
⊢ excenter t signs = incenter t

Tactic:
rcases h₁' with h₁' | h₁'

Goal state after:
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)
h₂' : excenter t signs = incenter t ∨ excenter t signs = excenter t {i₂}
h₁' : excenter t signs = incenter t
⊢ excenter t signs = incenter t

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
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
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 ∨ excenter t signs = excenter t {i₁}
h₂' : excenter t signs = excenter t {i₃} ∨ excenter t signs = excenter t {i₁}
⊢ excenter t signs = excenter t {i₁}

Tactic:
rcases h₁' with h₁' | h₁'

Goal state after:
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}

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) + ↑π
h₂' : excenter t signs = excenter t {i₃} ∨ excenter t signs = excenter t {i₁}
h₁' : excenter t signs = excenter t {i₁}
⊢ excenter t signs = excenter t {i₁}
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
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} ∨ excenter t signs = excenter t {i₃}
h₂' : excenter t signs = excenter t {i₃} ∨ excenter t signs = excenter t {i₁}
⊢ excenter t signs = excenter t {i₃}

Tactic:
rcases h₁' with h₁' | h₁'

Goal state after:
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}

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) + ↑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 {i₃}
⊢ excenter t signs = excenter t {i₃}
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
p₁ p₂ p₃ : P
s : Sphere P
inst✝ : Fact (finrank ℝ V = 2)
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
hp₃ : p₃ ∈ s
hne₁₂ : p₁ ≠ p₂
hne₂₃ : p₂ ≠ p₃
hangle : ∠ p₁ p₂ p₃ = π / 2
this : FiniteDimensional ℝ V
h : p₁ = p₃
⊢ False

Tactic:
rw [h, angle_self_of_ne hne₂₃.symm] at hangle; linarith [Real.pi_pos]

Goal state after:
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
⊢ 2 * (s.height i)⁻¹ < ∑ j ∈ {i}, (s.height j)⁻¹ + ∑ j ∈ {i}ᶜ, (s.height j)⁻¹

Tactic:
two_mul,

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 < ∑ j ∈ {i}, (s.height j)⁻¹ + ∑ j ∈ {i}ᶜ, (s.height j)⁻¹
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))
h : s.ExcenterExists signs
S : AffineSubspace ℝ P
hS : affineSpan ℝ (Set.range s.points) ≤ S
⊢ ↑((s.restrict S hS).excenter signs) = s.excenter signs

Tactic:
rw [← s.excenterExists_restrict S hS] at h
  have := Nonempty.map (AffineSubspace.inclusion hS) inferInstance
  exact (h.excenter_map S.subtypeₐᵢ).symm

Goal state after:
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))
h : s.ExcenterExists signs
i : Fin (n + 1)
⊢ (s.signedInfDist i) (s.excenter signs) = (if i ∈ signs then -1 else 1) * (∑ j, s.excenterWeightsUnnorm signs j)⁻¹

Tactic:
simp_rw [excenter_eq_affineCombination,
    signedInfDist_affineCombination _ _ h.sum_excenterWeights_eq_one, excenterWeights,
    Pi.smul_apply, ← dist_eq_norm_vsub, excenterWeightsUnnorm]
  rw [← altitudeFoot, ← height]
  simp [(s.height_pos i).ne']

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

Goal state before:
pp: case inl
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h0 : o.oangle x y ≠ 0
hpi : o.oangle x y ≠ ↑π
hs : (o.oangle x (x - y)).sign = (o.oangle (y - x) y).sign
h : angle x (x - y) = angle y (y - x)
⊢ ¬angle x y = π

Tactic:
← o.oangle_eq_pi_iff_angle_eq_pi

Goal state after:
case inl
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h0 : o.oangle x y ≠ 0
hpi : o.oangle x y ≠ ↑π
hs : (o.oangle x (x - y)).sign = (o.oangle (y - x) y).sign
h : angle x (x - y) = angle y (y - x)
⊢ ¬o.oangle x y = ↑π
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)
⊢ s.excenterWeights {i} i < 0

Tactic:
← sign_eq_neg_one_iff,

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)
⊢ SignType.sign (s.excenterWeights {i} i) = -1
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}
⊢ ¬Collinear ℝ {p₁, p₃, p₄}

Tactic:
rwa [← collinear_iff_of_two_zsmul_oangle_eq h]

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
x✝¹ x✝ : Fin (n + 1)
h : (fun i ↦ pointIndex i) x✝¹ = (fun i ↦ pointIndex i) x✝
⊢ x✝¹ = x✝

Tactic:
injection h

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

Goal state before:
pp: α : Type u_3
inst✝ : AddCommMonoid α
n : ℕ
f : PointsWithCircumcenterIndex n → α
h : univ = insert circumcenterIndex (Finset.map (pointIndexEmbedding n) univ)
⊢ ∑ i, f i = ∑ i, f ((pointIndexEmbedding n) i) + f circumcenterIndex

Tactic:
rw [add_comm, h, ← sum_map, sum_insert]

Goal state after:
α : Type u_3
inst✝ : AddCommMonoid α
n : ℕ
f : PointsWithCircumcenterIndex n → α
h : univ = insert circumcenterIndex (Finset.map (pointIndexEmbedding n) univ)
⊢ circumcenterIndex ∉ Finset.map (pointIndexEmbedding n) univ
Module: Mathlib.Geometry.Euclidean.Circumcenter

Goal state before:
pp: α : Type u_3
inst✝ : AddCommMonoid α
n : ℕ
f : PointsWithCircumcenterIndex n → α
h✝ : univ = insert circumcenterIndex (Finset.map (pointIndexEmbedding n) univ)
x : Fin (n + 1)
left✝ : x ∈ univ
h : (pointIndexEmbedding n) x = circumcenterIndex
⊢ False

Tactic:
injection h

Goal state after:
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
hc : Collinear ℝ {p₁, p₂, p₄}
he : ¬p₁ = p₄
⊢ Collinear ℝ {p₁, p₃, p₄}

Tactic:
rwa [← collinear_iff_of_two_zsmul_oangle_eq h]

Goal state after:
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
a b c : P
h : ¬Collinear ℝ {a, b, c}
hsin : Real.sin (∠ a b c) * dist a b = Real.sin (∠ a c b) * dist a c
hac : dist a c > 0
⊢ ∠ a b c ∈ Set.Icc 0 π

Tactic:
simp [angle_nonneg, angle_le_pi]

Goal state after:
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
a b c : P
h : ¬Collinear ℝ {a, b, c}
hsin : Real.sin (∠ a b c) * dist a b = Real.sin (∠ a c b) * dist a c
hac : dist a c > 0
hsinabc : Real.sin (∠ a b c) ≥ 0
h1 : ∠ a c b < ∠ a b c
⊢ dist a b < dist a c

Tactic:
by_cases! h2 : ∠ a b c ≤ π / 2

Goal state after:
case pos
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c : P
h : ¬Collinear ℝ {a, b, c}
hsin : Real.sin (∠ a b c) * dist a b = Real.sin (∠ a c b) * dist a c
hac : dist a c > 0
hsinabc : Real.sin (∠ a b c) ≥ 0
h1 : ∠ a c b < ∠ a b c
h2 : ∠ a b c ≤ π / 2
⊢ dist a b < dist a c

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
a b c : P
h : ¬Collinear ℝ {a, b, c}
hsin : Real.sin (∠ a b c) * dist a b = Real.sin (∠ a c b) * dist a c
hac : dist a c > 0
hsinabc : Real.sin (∠ a b c) ≥ 0
h1 : ∠ a c b < ∠ a b c
h2 : π / 2 < ∠ a b c
⊢ dist a b < dist a c
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.TriangleInequality

Goal state before:
pp: case right
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : ‖x‖ = 1
hy : ‖y‖ = 1
hxy : ⟪x, y⟫ = -1
⊢ x - ⟪x, y⟫ • y = 0

Tactic:
inner_eq_neg_one_iff_of_norm_eq_one hx hy

Goal state after:
case right
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : ‖x‖ = 1
hy : ‖y‖ = 1
hxy : x = -y
⊢ x - ⟪x, y⟫ • y = 0
Module: Mathlib.Geometry.Euclidean.Inversion.ImageHyperplane

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
c x y : P
R : ℝ
hR : R ≠ 0
hx : x ≠ c
hy : y ≠ c
⊢ dist (inversion c R x) c = dist (inversion c R x) (inversion c R y) ↔ dist x y = dist y c

Tactic:
dist_inversion_inversion hx hy,

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
c x y : P
R : ℝ
hR : R ≠ 0
hx : x ≠ c
hy : y ≠ c
⊢ dist (inversion c R x) c = R ^ 2 / (dist x c * dist y c) * dist x y ↔ dist x y = dist y c
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
hw : ∑ j, Simplex.touchpointWeights t {i₁} i₂ j = 1
⊢ Sbtw ℝ ((Finset.affineCombination ℝ Finset.univ t.points) (Simplex.touchpointWeights t {i₁} i₂)) (t.points i₃)
    (t.points i₁)

Tactic:
rw [(by clear hw; decide +revert : (Finset.univ : Finset (Fin 3)) = {i₁, i₂, i₃})] at hw

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
hw : ∑ j ∈ {i₁, i₂, i₃}, Simplex.touchpointWeights t {i₁} i₂ j = 1
⊢ Sbtw ℝ ((Finset.affineCombination ℝ Finset.univ t.points) (Simplex.touchpointWeights t {i₁} i₂)) (t.points i₃)
    (t.points i₁)
Module: Mathlib.Geometry.Euclidean.Incenter

Goal state before:
pp: case inr.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
t : Triangle ℝ P
i₁ i₃ : Fin 3
h₁₃ : i₁ ≠ i₃
i : Fin (2 + 1)
h₁₂ : i₁ ≠ i
h₂₃ : i ≠ i₃
hw : Simplex.touchpointWeights t {i₁} i i₁ + Simplex.touchpointWeights t {i₁} i i₃ = 1
⊢ Simplex.touchpointWeights t {i₁} i i =
    Finset.affineCombinationLineMapWeights i₁ i₃ (Simplex.touchpointWeights t {i₁} i i₃) i

Tactic:
simp [h₁₂.symm, h₂₃]

Goal state after: