awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Geometry.Euclidean.Angle.Incenter
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst_4 : 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)
h : ∡ (t.points i₂) (t.points i₁) (excenter t signs) = ∡ (excenter t signs) (t.points i₁) (t.points i₃)
hs : excenter t signs = excenter t {i₂}
⊢ ∡ (excenter t {i₂}) (t.points i₁) (t.points i₃) + (↑π : Real.Angle) = ∡ (excenter t {i₂}) (t.points i₁) (t.points i₃)

Tactic:
rw [hs, t.oangle_excenter_singleton_eq_add_pi h₁₂.symm h₂₃ h₁₃] at h

Premises used: Affine.Triangle.oangle_excenter_singleton_eq_add_pi, Ne.symm
-- Module: Mathlib.Geometry.Euclidean.Angle.Incenter
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst_4 : 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)
h : ∡ (t.points i₂) (t.points i₁) (excenter t signs) = ∡ (excenter t signs) (t.points i₁) (t.points i₃)
hs : excenter t signs = excenter t {i₂}
h₁ :
  ∡ (excenter t {i₂}) (t.points i₁) (t.points i₃) + (↑π : Real.Angle) = ∡ (excenter t {i₂}) (t.points i₁) (t.points i₃)
⊢ excenter t signs = excenter t {i₂}

Tactic:
rw [hs, t.oangle_excenter_singleton_eq_add_pi h₁₂.symm h₂₃ h₁₃] at h

Premises used: Affine.Triangle.oangle_excenter_singleton_eq_add_pi, Ne.symm
-- Module: Mathlib.Geometry.Euclidean.Angle.Incenter
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst_4 : 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)
h :
  ∡ (excenter t {i₂}) (t.points i₁) (t.points i₃) + (↑π : Real.Angle) = ∡ (excenter t {i₂}) (t.points i₁) (t.points i₃)
hs : excenter t signs = excenter t {i₂}
⊢ excenter t signs = incenter t ∨ excenter t signs = excenter t {i₁}

Tactic:
simp [Real.Angle.pi_ne_zero] at h

Premises used: Real.Angle.pi_ne_zero
-- Module: Mathlib.Geometry.Euclidean.Angle.Incenter
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst_4 : 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)
h : ∡ (t.points i₂) (t.points i₁) (excenter t signs) = ∡ (excenter t signs) (t.points i₁) (t.points i₃)
hs : excenter t signs = excenter t {i₃}
⊢ -∡ (excenter t {i₃}) (t.points i₁) (t.points i₂) =
    -(∡ (excenter t {i₃}) (t.points i₁) (t.points i₂) + (↑π : Real.Angle))

Tactic:
rw [hs, oangle_rev (t.points i₃), t.oangle_excenter_singleton_eq_add_pi h₁₃.symm h₂₃.symm h₁₂,
      oangle_rev] at h

Premises used: Affine.Triangle.oangle_excenter_singleton_eq_add_pi, Affine.Simplex.points, EuclideanGeometry.oangle_rev, Ne.symm
-- Module: Mathlib.Geometry.Euclidean.Angle.Incenter
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst_4 : 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)
h : ∡ (t.points i₂) (t.points i₁) (excenter t signs) = ∡ (excenter t signs) (t.points i₁) (t.points i₃)
hs : excenter t signs = excenter t {i₃}
h₁ :
  -∡ (excenter t {i₃}) (t.points i₁) (t.points i₂) =
    -(∡ (excenter t {i₃}) (t.points i₁) (t.points i₂) + (↑π : Real.Angle))
⊢ excenter t signs = excenter t {i₃}

Tactic:
rw [hs, oangle_rev (t.points i₃), t.oangle_excenter_singleton_eq_add_pi h₁₃.symm h₂₃.symm h₁₂,
      oangle_rev] at h

Premises used: Affine.Triangle.oangle_excenter_singleton_eq_add_pi, Affine.Simplex.points, EuclideanGeometry.oangle_rev, Ne.symm
-- Module: Mathlib.Geometry.Euclidean.Angle.Incenter
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst_4 : 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)
h :
  -∡ (excenter t {i₃}) (t.points i₁) (t.points i₂) =
    -(∡ (excenter t {i₃}) (t.points i₁) (t.points i₂) + (↑π : Real.Angle))
hs : excenter t signs = excenter t {i₃}
⊢ excenter t signs = incenter t ∨ excenter t signs = excenter t {i₁}

Tactic:
simp [Real.Angle.pi_ne_zero] at h

Premises used: Real.Angle.pi_ne_zero
-- Module: Mathlib.Geometry.Euclidean.Angle.Incenter
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst_4 : 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)
h :
  ∡ (t.points i₂) (t.points i₁) (excenter t signs) =
    ∡ (excenter t signs) (t.points i₁) (t.points i₃) + (↑π : Real.Angle)
⊢ excenter t signs = incenter t ∨
    excenter t signs = excenter t {i₁} ∨ excenter t signs = excenter t {i₂} ∨ excenter t signs = excenter t {i₃}

Tactic:
have hs := t.excenter_eq_incenter_or_excenter_singleton_of_ne signs h₁₂ h₁₃ h₂₃
-- Module: Mathlib.Geometry.Euclidean.Angle.Incenter
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst_4 : 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)
h :
  ∡ (t.points i₂) (t.points i₁) (excenter t signs) =
    ∡ (excenter t signs) (t.points i₁) (t.points i₃) + (↑π : Real.Angle)
hs :
  excenter t signs = incenter t ∨
    excenter t signs = excenter t {i₁} ∨ excenter t signs = excenter t {i₂} ∨ excenter t signs = excenter t {i₃}
inl : ∀ (hs : excenter t signs = incenter t), excenter t signs = excenter t {i₂} ∨ excenter t signs = excenter t {i₃}
inr :
  ∀ (hs : excenter t signs = excenter t {i₁}), excenter t signs = excenter t {i₂} ∨ excenter t signs = excenter t {i₃}
inr₁ :
  ∀ (hs : excenter t signs = excenter t {i₂}), excenter t signs = excenter t {i₂} ∨ excenter t signs = excenter t {i₃}
inr₂ :
  ∀ (hs : excenter t signs = excenter t {i₃}), excenter t signs = excenter t {i₂} ∨ excenter t signs = excenter t {i₃}
⊢ excenter t signs = excenter t {i₂} ∨ excenter t signs = excenter t {i₃}

Tactic:
rcases hs with hs | hs | hs | hs
-- Module: Mathlib.Geometry.Euclidean.Angle.Incenter
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst_4 : 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)
h :
  ∡ (t.points i₂) (t.points i₁) (excenter t signs) =
    ∡ (excenter t signs) (t.points i₁) (t.points i₃) + (↑π : Real.Angle)
hs : excenter t signs = incenter t
⊢ ∡ (incenter t) (t.points i₁) (t.points i₃) = ∡ (incenter t) (t.points i₁) (t.points i₃) + (↑π : Real.Angle)

Tactic:
rw [hs, t.oangle_incenter_eq h₁₂ h₁₃ h₂₃] at h

Premises used: Affine.Triangle.oangle_incenter_eq
-- Module: Mathlib.Geometry.Euclidean.Angle.Incenter
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst_4 : 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)
h :
  ∡ (t.points i₂) (t.points i₁) (excenter t signs) =
    ∡ (excenter t signs) (t.points i₁) (t.points i₃) + (↑π : Real.Angle)
hs : excenter t signs = incenter t
h₁ : ∡ (incenter t) (t.points i₁) (t.points i₃) = ∡ (incenter t) (t.points i₁) (t.points i₃) + (↑π : Real.Angle)
⊢ excenter t signs = incenter t

Tactic:
rw [hs, t.oangle_incenter_eq h₁₂ h₁₃ h₂₃] at h

Premises used: Affine.Triangle.oangle_incenter_eq
-- Module: Mathlib.Geometry.Euclidean.Angle.Incenter
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst_4 : 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)
h : ∡ (incenter t) (t.points i₁) (t.points i₃) = ∡ (incenter t) (t.points i₁) (t.points i₃) + (↑π : Real.Angle)
hs : excenter t signs = incenter t
⊢ excenter t signs = excenter t {i₂} ∨ excenter t signs = excenter t {i₃}

Tactic:
simp [Real.Angle.pi_ne_zero] at h

Premises used: Real.Angle.pi_ne_zero
-- Module: Mathlib.Geometry.Euclidean.Angle.Incenter
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst_4 : 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)
h :
  ∡ (t.points i₂) (t.points i₁) (excenter t signs) =
    ∡ (excenter t signs) (t.points i₁) (t.points i₃) + (↑π : Real.Angle)
hs : excenter t signs = excenter t {i₁}
⊢ ∡ (excenter t {i₁}) (t.points i₁) (t.points i₃) = ∡ (excenter t {i₁}) (t.points i₁) (t.points i₃) + (↑π : Real.Angle)

Tactic:
rw [hs, t.oangle_excenter_singleton_eq h₁₂ h₁₃ h₂₃] at h

Premises used: Affine.Triangle.oangle_excenter_singleton_eq
-- Module: Mathlib.Geometry.Euclidean.Angle.Incenter
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst_4 : 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)
h :
  ∡ (t.points i₂) (t.points i₁) (excenter t signs) =
    ∡ (excenter t signs) (t.points i₁) (t.points i₃) + (↑π : Real.Angle)
hs : excenter t signs = excenter t {i₁}
h₁ :
  ∡ (excenter t {i₁}) (t.points i₁) (t.points i₃) = ∡ (excenter t {i₁}) (t.points i₁) (t.points i₃) + (↑π : Real.Angle)
⊢ excenter t signs = excenter t {i₁}

Tactic:
rw [hs, t.oangle_excenter_singleton_eq h₁₂ h₁₃ h₂₃] at h

Premises used: Affine.Triangle.oangle_excenter_singleton_eq
-- Module: Mathlib.Geometry.Euclidean.Angle.Incenter
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst_4 : 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)
h :
  ∡ (excenter t {i₁}) (t.points i₁) (t.points i₃) = ∡ (excenter t {i₁}) (t.points i₁) (t.points i₃) + (↑π : Real.Angle)
hs : excenter t signs = excenter t {i₁}
⊢ excenter t signs = excenter t {i₂} ∨ excenter t signs = excenter t {i₃}

Tactic:
simp [Real.Angle.pi_ne_zero] at h

Premises used: Real.Angle.pi_ne_zero
-- Module: Mathlib.Geometry.Euclidean.Angle.Incenter
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst_4 : 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)
h :
  ∡ (t.points i₂) (t.points i₁) (excenter t signs) =
    ∡ (excenter t signs) (t.points i₁) (t.points i₃) + (↑π : Real.Angle)
hs : excenter t signs = excenter t {i₂}
⊢ excenter t signs = excenter t {i₂} ∨ excenter t signs = excenter t {i₃}

Tactic:
.inl hs

Premises used: Or.inl
-- Module: Mathlib.Geometry.Euclidean.Angle.Incenter
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst_4 : 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)
h :
  ∡ (t.points i₂) (t.points i₁) (excenter t signs) =
    ∡ (excenter t signs) (t.points i₁) (t.points i₃) + (↑π : Real.Angle)
hs : excenter t signs = excenter t {i₂}
⊢ excenter t signs = excenter t {i₂} ∨ excenter t signs = excenter t {i₃}

Tactic:
exact .inl hs
-- Module: Mathlib.Geometry.Euclidean.Angle.Incenter
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst_4 : 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)
h :
  ∡ (t.points i₂) (t.points i₁) (excenter t signs) =
    ∡ (excenter t signs) (t.points i₁) (t.points i₃) + (↑π : Real.Angle)
hs : excenter t signs = excenter t {i₃}
⊢ excenter t signs = excenter t {i₂} ∨ excenter t signs = excenter t {i₃}

Tactic:
.inr hs

Premises used: Or.inr
-- Module: Mathlib.Geometry.Euclidean.Angle.Incenter
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst_4 : 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)
h :
  ∡ (t.points i₂) (t.points i₁) (excenter t signs) =
    ∡ (excenter t signs) (t.points i₁) (t.points i₃) + (↑π : Real.Angle)
hs : excenter t signs = excenter t {i₃}
⊢ excenter t signs = excenter t {i₂} ∨ excenter t signs = excenter t {i₃}

Tactic:
exact .inr hs
-- Module: Mathlib.Geometry.Euclidean.Angle.Incenter
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst_4 : 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 : ∡ (t.points i₂) (t.points i₁) p = ∡ p (t.points i₁) (t.points i₃)
h₂ : ∡ (t.points i₃) (t.points i₂) p = ∡ p (t.points i₂) (t.points i₁)
rw rw₁ :
  HSMul.hSMul (α := ℤ) 2 (∡ p (t.points i₁) (t.points i₃)) = HSMul.hSMul (α := ℤ) 2 (∡ p (t.points i₁) (t.points i₃))
⊢ HSMul.hSMul (α := ℤ) 2 (∡ (t.points i₂) (t.points i₁) p) = HSMul.hSMul (α := ℤ) 2 (∡ p (t.points i₁) (t.points i₃))

Tactic:
rw [h₁]
-- Module: Mathlib.Geometry.Euclidean.Angle.Incenter
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst_4 : 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 : ∡ (t.points i₂) (t.points i₁) p = ∡ p (t.points i₁) (t.points i₃)
h₂ : ∡ (t.points i₃) (t.points i₂) p = ∡ p (t.points i₂) (t.points i₁)
rw rw₁ :
  HSMul.hSMul (α := ℤ) 2 (∡ p (t.points i₂) (t.points i₁)) = HSMul.hSMul (α := ℤ) 2 (∡ p (t.points i₂) (t.points i₁))
⊢ HSMul.hSMul (α := ℤ) 2 (∡ (t.points i₃) (t.points i₂) p) = HSMul.hSMul (α := ℤ) 2 (∡ p (t.points i₂) (t.points i₁))

Tactic:
rw [h₂]
-- Module: Mathlib.Geometry.Euclidean.Angle.Incenter
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst_4 : 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 : ∡ (t.points i₂) (t.points i₁) p = ∡ p (t.points i₁) (t.points i₃)
h₂ : ∡ (t.points i₃) (t.points i₂) p = ∡ p (t.points i₂) (t.points i₁)
obtain :
  ∀ (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)),
    excenter t signs = incenter t
⊢ p = incenter t

Tactic:
obtain ⟨signs, rfl⟩ := t.eq_excenter_of_two_zsmul_oangle_eq h₁₂ h₁₃ h₂₃ (by rw [h₁]) (by rw [h₂])

Premises used: Affine.Triangle.eq_excenter_of_two_zsmul_oangle_eq, rfl
-- Module: Mathlib.Geometry.Euclidean.Angle.Incenter
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst_4 : 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)
⊢ excenter t signs = incenter t ∨ excenter t signs = excenter t {i₁}

Tactic:
have h₁' := t.eq_incenter_or_eq_excenter_singleton_of_oangle_eq h₁₂ h₁₃ h₂₃ h₁
-- Module: Mathlib.Geometry.Euclidean.Angle.Incenter
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst_4 : 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}
⊢ excenter t signs = incenter t ∨ excenter t signs = excenter t {i₂}

Tactic:
have h₂' := t.eq_incenter_or_eq_excenter_singleton_of_oangle_eq h₂₃ h₁₂.symm h₁₃.symm h₂
-- Module: Mathlib.Geometry.Euclidean.Angle.Incenter
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst_4 : 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}
inl : ∀ (h₁' : excenter t signs = incenter t), excenter t signs = incenter t
inr : ∀ (h₁' : excenter t signs = excenter t {i₁}), excenter t signs = incenter t
⊢ excenter t signs = incenter t

Tactic:
rcases h₁' with h₁' | h₁'
-- Module: Mathlib.Geometry.Euclidean.Angle.Incenter
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst_4 : 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

Tactic:
h₁'
-- Module: Mathlib.Geometry.Euclidean.Angle.Incenter
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst_4 : 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

Tactic:
exact h₁'
-- Module: Mathlib.Geometry.Euclidean.Angle.Incenter
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst_4 : 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}
inr : ∀ (h₂' : excenter t signs = incenter t), excenter t signs = incenter t
inr₁ : ∀ (h₂' : excenter t signs = excenter t {i₂}), excenter t signs = incenter t
⊢ excenter t signs = incenter t

Tactic:
rcases h₂' with h₂' | h₂'
-- Module: Mathlib.Geometry.Euclidean.Angle.Incenter
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst_4 : 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

Tactic:
h₂'
-- Module: Mathlib.Geometry.Euclidean.Angle.Incenter
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst_4 : 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

Tactic:
exact h₂'
-- Module: Mathlib.Geometry.Euclidean.Angle.Incenter
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst_4 : 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 {i₁} = excenter t {i₂}

Tactic:
rw [h₁'] at h₂'
-- Module: Mathlib.Geometry.Euclidean.Angle.Incenter
Goal:
V : Type u_1
P : Type u_2
inst : NormedAddCommGroup V
inst_1 : InnerProductSpace ℝ V
inst_2 : MetricSpace P
inst_3 : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst_4 : 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 {i₁} = excenter t {i₂}
inr : False
⊢ excenter t signs = incenter t

Tactic:
exfalso