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

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 : Simplex ℝ P 1
i : Fin 2
⊢ s.height i = dist (s.points 0) (s.points 1)

Tactic:
by
  fin_cases i
  · simp [height]
  · rw [dist_comm]
    simp [height]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation

Goal state before:
pp: 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 : V
θ : Real.Angle
⊢ (o.rotation θ) x = x ↔ x = 0 ∨ θ = 0

Tactic:
by_cases h : x = 0

Goal state after:
case pos
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 : V
θ : Real.Angle
h : x = 0
⊢ (o.rotation θ) x = x ↔ x = 0 ∨ θ = 0

case neg
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 : V
θ : Real.Angle
h : ¬x = 0
⊢ (o.rotation θ) x = x ↔ x = 0 ∨ θ = 0
Module: Mathlib.Geometry.Euclidean.Projection

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace 𝕜 V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex 𝕜 P n
S : AffineSubspace 𝕜 P
hS : affineSpan 𝕜 (Set.range s.points) ≤ S
p : ↥S
⊢ ↑(s.orthogonalProjectionSpan ↑p) = ↑↑((s.restrict S hS).orthogonalProjectionSpan p)

Tactic:
convert! (s.restrict S hS).orthogonalProjectionSpan_map S.subtypeₐᵢ p

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation

Goal state before:
pp: 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
hx : x ≠ 0
hy : y ≠ 0
θ : Real.Angle
⊢ o.oangle x y = θ ↔ y = (‖y‖ / ‖x‖) • (o.rotation θ) x

Tactic:
have hp := div_pos (norm_pos_iff.2 hy) (norm_pos_iff.2 hx)
  constructor
  · rintro rfl
    rw [← map_smul, ← o.oangle_smul_left_of_pos x y hp, eq_comm,
      rotation_oangle_eq_iff_norm_eq, norm_smul, Real.norm_of_nonneg hp.le,
      div_mul_cancel₀ _ (norm_ne_zero_iff.2 hx)]
  · intro hye
    rw [hye, o.oangle_smul_right_of_pos _ _ hp, o.oangle_rotation_self_right hx]

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

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
x : P × P × P
hx12 : x.1 ≠ x.2.1
hx32 : x.2.2 ≠ x.2.1
f : P × P × P → V × V := fun y ↦ (y.1 -ᵥ y.2.1, y.2.2 -ᵥ y.2.1)
hf1 : (f x).1 ≠ 0
⊢ (f x).2 ≠ 0

Tactic:
simp [f, hx32]

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

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
s : Simplex ℝ P n
inst✝ : n.AtLeastTwo
i j : Fin (n + 1)
hij : i ≠ j
⊢ 0 ≤ s.height i * s.height j

Tactic:
simp only [height]
    positivity

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

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
⊢ ∠ p₁ p₂ p₃ = 0 ↔ p₁ ≠ p₂ ∧ Wbtw ℝ p₂ p₁ p₃ ∨ p₃ ≠ p₂ ∧ Wbtw ℝ p₂ p₃ p₁

Tactic:
constructor
  · rw [angle, angle_eq_zero_iff]
    rintro ⟨hp₁p₂, r, hr0, hp₃p₂⟩
    rcases le_or_gt 1 r with (hr1 | hr1)
    · refine Or.inl ⟨vsub_ne_zero.1 hp₁p₂, r⁻¹, ⟨(inv_pos.2 hr0).le, inv_le_one_of_one_le₀ hr1⟩, ?_⟩
      rw [AffineMap.lineMap_apply, hp₃p₂, smul_smul, inv_mul_cancel₀ hr0.ne.symm, one_smul,
        vsub_vadd]
    · refine Or.inr ⟨?_, r, ⟨hr0.le, hr1.le⟩, ?_⟩
      · rw [← @vsub_ne_zero V, hp₃p₂, smul_ne_zero_iff]
        exact ⟨hr0.ne.symm, hp₁p₂⟩
      · rw [AffineMap.lineMap_apply, ← hp₃p₂, vsub_vadd]
  · rintro (⟨hp₁p₂, h⟩ | ⟨hp₃p₂, h⟩)
    · exact h.angle₂₁₃_eq_zero_of_ne hp₁p₂
    · exact h.angle₃₁₂_eq_zero_of_ne hp₃p₂

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

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫ = 0
⊢ Real.cos (angle x (x - y)) = ‖x‖ / ‖x - y‖

Tactic:
rw [← neg_eq_zero, ← inner_neg_right] at h
  rw [sub_eq_add_neg, cos_angle_add_of_inner_eq_zero h]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Basic

Goal state before:
pp: case neg
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)
w x y z : V
h : |(o.oangle w x).toReal| = InnerProductGeometry.angle y z
hs : (o.oangle w x).sign = (o.oangle y z).sign
h0 : (w ≠ 0 ∧ x ≠ 0) ∧ y ≠ 0 ∧ z ≠ 0
⊢ |(o.oangle w x).toReal| = |(o.oangle y z).toReal|

Tactic:
o.angle_eq_abs_oangle_toReal h0.2.1 h0.2.2

Goal state after:
case neg
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)
w x y z : V
h : |(o.oangle w x).toReal| = |(o.oangle y z).toReal|
hs : (o.oangle w x).sign = (o.oangle y z).sign
h0 : (w ≠ 0 ∧ x ≠ 0) ∧ y ≠ 0 ∧ z ≠ 0
⊢ |(o.oangle w x).toReal| = |(o.oangle y z).toReal|
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = ↑(π / 2)
⊢ ‖y‖ / ((-o).oangle x (x - y)).sin = ‖x - y‖

Tactic:
exact (-o).norm_div_sin_oangle_sub_right_of_oangle_eq_pi_div_two h

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle

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 : P
h : ∡ p₁ p₂ p₃ = ↑(π / 2)
⊢ ∡ p₃ p₁ p₂ = ↑(Real.arctan (dist p₃ p₂ / dist p₁ p₂))

Tactic:
by
  have hs : (∡ p₃ p₁ p₂).sign = 1 := by rw [← oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]
  rw [oangle_eq_angle_of_sign_eq_one hs, angle_comm,
    angle_eq_arctan_of_angle_eq_pi_div_two (angle_rev_eq_pi_div_two_of_oangle_eq_pi_div_two h)
      (left_ne_of_oangle_eq_pi_div_two h)]

Goal state after:
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
p p' : P
s₁ s₂ : AffineSubspace ℝ P
inst✝¹ : s₁.direction.HasOrthogonalProjection
inst✝ : s₂.direction.HasOrthogonalProjection
hp'₁ : p' ∈ s₁
hp'₂ : p' ∈ s₂
this✝ : Nonempty ↥s₁
this : Nonempty ↥s₂
⊢ dist p ↑((orthogonalProjection s₁) p) = dist p ↑((orthogonalProjection s₂) p) ↔
    ∠ p p' ↑((orthogonalProjection s₁) p) = ∠ p p' ↑((orthogonalProjection s₂) p)

Tactic:
by_cases h' : p ∈ s₁ ∨ p ∈ s₂

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
p p' : P
s₁ s₂ : AffineSubspace ℝ P
inst✝¹ : s₁.direction.HasOrthogonalProjection
inst✝ : s₂.direction.HasOrthogonalProjection
hp'₁ : p' ∈ s₁
hp'₂ : p' ∈ s₂
this✝ : Nonempty ↥s₁
this : Nonempty ↥s₂
h' : p ∈ s₁ ∨ p ∈ s₂
⊢ dist p ↑((orthogonalProjection s₁) p) = dist p ↑((orthogonalProjection s₂) p) ↔
    ∠ p p' ↑((orthogonalProjection s₁) p) = ∠ p p' ↑((orthogonalProjection s₂) p)

case neg
V : Type u_1
P : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
p p' : P
s₁ s₂ : AffineSubspace ℝ P
inst✝¹ : s₁.direction.HasOrthogonalProjection
inst✝ : s₂.direction.HasOrthogonalProjection
hp'₁ : p' ∈ s₁
hp'₂ : p' ∈ s₂
this✝ : Nonempty ↥s₁
this : Nonempty ↥s₂
h' : ¬(p ∈ s₁ ∨ p ∈ s₂)
⊢ dist p ↑((orthogonalProjection s₁) p) = dist p ↑((orthogonalProjection s₂) p) ↔
    ∠ p p' ↑((orthogonalProjection s₁) p) = ∠ p p' ↑((orthogonalProjection s₂) p)
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Basic

Goal state before:
pp: case pos
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 z : V
hx : x ≠ 0
hz : z ≠ 0
hy : ¬y = 0
hr : SameRay ℝ x z
⊢ InnerProductGeometry.angle x y = InnerProductGeometry.angle y z ↔ o.oangle x y = o.oangle y z ∨ SameRay ℝ x z

Tactic:
obtain ⟨r, hrp, rfl⟩ := hr.exists_pos_left hx hz
    simp [hr, hrp, InnerProductGeometry.angle_comm]

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

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 p₁ p₂ : P
⊢ ⟪c -ᵥ midpoint ℝ p₁ ((Equiv.pointReflection p₂) p₁), (Equiv.pointReflection p₂) p₁ -ᵥ p₁⟫ = 0 ↔
    ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ = 0

Tactic:
midpoint_pointReflection_right,

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 p₁ p₂ : P
⊢ ⟪c -ᵥ p₂, (Equiv.pointReflection p₂) p₁ -ᵥ p₁⟫ = 0 ↔ ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ = 0
Module: Mathlib.Geometry.Euclidean.SignedDist

Goal state before:
pp: case neg
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
v : V
p q : P
h : ¬v = 0
⊢ |⟪NormedSpace.normalize v, q -ᵥ p⟫| ≤ dist p q

Tactic:
abs_real_inner_le_norm

Goal state after:
case neg
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
v : V
p q : P
h : ¬v = 0
⊢ ‖NormedSpace.normalize v‖ * ‖q -ᵥ p‖ ≤ dist p q
Module: Mathlib.Geometry.Euclidean.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
ι : Type u_3
s : Finset ι
w₁ w₂ : ι → ℝ
p : ι → P
h₁ : ∑ i ∈ s, w₁ i = 1
h₂ : ∑ i ∈ s, w₂ i = 1
⊢ Sort ?u.41

Tactic:
have a₁ := s.affineCombination ℝ p w₁
      have a₂ := s.affineCombination ℝ p w₂
      exact dist a₁ a₂ * dist a₁ a₂ = (-∑ i₁ ∈ s, ∑ i₂ ∈ s,
        (w₁ - w₂) i₁ * (w₁ - w₂) i₂ * (dist (p i₁) (p i₂) * dist (p i₁) (p i₂))) / 2

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Affine

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
h₁₂ : p₁ ≠ p₂
ha : ∡ p₁ p₂ p₃ = ∡ p₃ p₂ p₄
hs : line[ℝ, p₁, p₂].SSameSide p₃ p₄
h₃₂ : p₃ ≠ p₂
h₄₂ : p₄ ≠ p₂
⊢ ∠ p₁ p₂ p₃ = ∠ p₁ p₂ p₄ / 2

Tactic:
suffices ((∡ p₁ p₂ p₃).toReal + (∡ p₃ p₂ p₄).toReal) / 2 = (∡ p₁ p₂ p₄).toReal / 2 by
    rw [← ha, add_self_div_two] at this
    rw [angle_eq_abs_oangle_toReal h₁₂ h₃₂, angle_eq_abs_oangle_toReal h₁₂ h₄₂, this, abs_div]
    simp

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
h₁₂ : p₁ ≠ p₂
ha : ∡ p₁ p₂ p₃ = ∡ p₃ p₂ p₄
hs : line[ℝ, p₁, p₂].SSameSide p₃ p₄
h₃₂ : p₃ ≠ p₂
h₄₂ : p₄ ≠ p₂
⊢ ((∡ p₁ p₂ p₃).toReal + (∡ p₃ p₂ p₄).toReal) / 2 = (∡ p₁ p₂ p₄).toReal / 2
Module: Mathlib.Geometry.Euclidean.Sphere.Basic

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : InnerProductSpace ℝ V
inst✝⁴ : MetricSpace P
inst✝³ : NormedAddTorsor V P
S₁ S₂ : AffineSubspace ℝ P
inst✝² : Nonempty ↥S₁
ps : Set ↥S₁
inst✝¹ : S₁.direction.HasOrthogonalProjection
inst✝ : S₂.direction.HasOrthogonalProjection
hS : S₁ ≤ S₂
⊢ Cospherical (⇑(AffineSubspace.inclusion hS) '' ps) ↔ Cospherical ps

Tactic:
have : Nonempty S₂ := by obtain ⟨p⟩ := ‹Nonempty S₁›; exact ⟨⟨p, hS p.property⟩⟩
  simp [(Cospherical.subtype_val_iff (S := S₂) (ps := AffineSubspace.inclusion hS '' ps)).symm,
    Set.image_image]

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

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 : 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
⊢ p = p₁ ∨ p = p₂

Tactic:
have hbs : Submodule.span ℝ (Set.range b) = s.direction := by
    refine Submodule.eq_of_le_of_finrank_eq ?_ ?_
    · 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
    · rw [finrank_span_eq_card hb, Fintype.card_fin, hd]

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 : 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
hbs : Submodule.span ℝ (Set.range b) = s.direction
⊢ p = p₁ ∨ p = p₂
Module: Mathlib.Geometry.Euclidean.Angle.Bisector

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
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p p₁ p₂ p₃ : P
h₂ : p₁ ≠ p₂
h : 2 • ∡ p₂ p₁ p = 2 • ∡ p p₁ p₃
h' : ↑((orthogonalProjection line[ℝ, p₁, p₂]) p) = p₁
hp : p ≠ p₁
⊢ dist p ↑((orthogonalProjection line[ℝ, p₁, p₂]) p) = dist p ↑((orthogonalProjection line[ℝ, p₁, p₃]) p)

Tactic:
· rw [← h'] at h hp
    have hpm : p ∉ line[ℝ, p₁, p₂] := orthogonalProjection_eq_self_iff.not.1 (Ne.symm hp)
    rw [two_zsmul_oangle_orthogonalProjection_self _ hpm (right_mem_affineSpan_pair _ _ _)
          (h'.symm ▸ h₂.symm), eq_comm, oangle, Real.Angle.two_zsmul_eq_pi_iff, h'] at h
    replace h := (Orientation.eq_zero_or_oangle_eq_iff_inner_eq_zero _).1 (.inr (.inr h))
    congr 1
    rw [h', eq_comm, coe_orthogonalProjection_eq_iff_mem]
    refine ⟨left_mem_affineSpan_pair _ _ _, ?_⟩
    rw [Submodule.mem_orthogonal']
    intro u hu
    rw [direction_affineSpan, mem_vectorSpan_pair] at hu
    rcases hu with ⟨r, rfl⟩
    rw [inner_smul_right, ← inner_neg_neg, inner_neg_left]
    simp [h]

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
p₁ p₂ : P
s : Sphere P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
hp₁p₂ : p₁ ≠ p₂
hp₁' : ‖p₁ -ᵥ s.center‖ = s.radius
hp₂' : ‖p₂ -ᵥ s.center‖ = s.radius
hd : s.radius ^ 2 = ‖p₂ -ᵥ p₁‖ ^ 2 + 2 * ⟪p₂ -ᵥ p₁, p₁ -ᵥ s.center⟫ + s.radius ^ 2
⊢ 0 < ⟪p₂ -ᵥ p₁, s.center -ᵥ p₁⟫

Tactic:
← neg_vsub_eq_vsub_rev s.center p₁,

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
s : Sphere P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
hp₁p₂ : p₁ ≠ p₂
hp₁' : ‖p₁ -ᵥ s.center‖ = s.radius
hp₂' : ‖p₂ -ᵥ s.center‖ = s.radius
hd : s.radius ^ 2 = ‖p₂ -ᵥ p₁‖ ^ 2 + 2 * ⟪p₂ -ᵥ p₁, -(s.center -ᵥ p₁)⟫ + s.radius ^ 2
⊢ 0 < ⟪p₂ -ᵥ p₁, s.center -ᵥ p₁⟫
Module: Mathlib.Geometry.Euclidean.Sphere.OrthRadius

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
h : dist p s.center < s.radius
hp : p ≠ s.center
hb : ℝ ∙ (p -ᵥ s.center) ≠ ⊤
⊢ ¬Subsingleton P

Tactic:
intro hs

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
h : dist p s.center < s.radius
hp : p ≠ s.center
hb : ℝ ∙ (p -ᵥ s.center) ≠ ⊤
hs : Subsingleton P
⊢ False
Module: Mathlib.Geometry.Euclidean.Sphere.OrthRadius

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
s : Sphere P
p : P
h : dist p s.center = s.radius
⊢ Subsingleton V → s.radius = 0

Tactic:
intro hs

Goal state after:
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
s : Sphere P
p : P
h : dist p s.center = s.radius
hs : Subsingleton V
⊢ s.radius = 0
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
⊢ ∡ (touchpoint t {i₁} i₃) (t.points i₂) (excenter t {i₁}) = ∡ (excenter t {i₁}) (t.points i₂) (touchpoint t {i₁} i₁)

Tactic:
have hd := (t.excenterExists_singleton i₁).dist_excenter_eq_dist_excenter i₃ i₁

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
hd : dist (excenter t {i₁}) (touchpoint t {i₁} i₃) = dist (excenter t {i₁}) (touchpoint t {i₁} i₁)
⊢ ∡ (touchpoint t {i₁} i₃) (t.points i₂) (excenter t {i₁}) = ∡ (excenter t {i₁}) (t.points i₂) (touchpoint t {i₁} i₁)
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}

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) + ↑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}

Tactic:
· rcases h₂' with h₂' | h₂'
    · exact h₂'
    rw [h₂'] at h₁'
    exfalso
    exact t.excenter_singleton_injective.ne h₁₂ h₁'

Goal state after:
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_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
⊢ ∡ p₁ s.center p₂ = ↑π - 2 • ∡ s.center p₂ p₁

Tactic:
by
  rw [oangle_eq_pi_sub_two_zsmul_oangle_of_dist_eq h.symm
      (dist_center_eq_dist_center_of_mem_sphere' hp₂ hp₁)]

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
r : ℝ
s : Simplex ℝ P n
p₁ : P
h₁ : ∀ (i : Fin (n + 1)), dist (s.points i) p₁ = r
h₁' : ↑(s.orthogonalProjectionSpan p₁) = s.circumcenter
h : s.points 0 ∈ affineSpan ℝ (Set.range s.points)
⊢ dist p₁ s.circumcenter * dist p₁ s.circumcenter = r * r - s.circumradius * s.circumradius

Tactic:
by
  rw [dist_comm, ← h₁ 0,
    s.dist_sq_eq_dist_orthogonalProjection_sq_add_dist_orthogonalProjection_sq p₁ h]
  simp only [h₁', dist_comm p₁, add_sub_cancel_left, Simplex.dist_circumcenter_eq_circumradius]

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

Goal state before:
pp: case e'_2.a
n : ℕ
j : PointsWithCircumcenterIndex n
a✝ : j ∈ univ
⊢ circumcenterWeightsWithCircumcenter n j =
    if j = circumcenterIndex then Function.const (PointsWithCircumcenterIndex n) 1 j else 0

pp: case e'_3
n : ℕ
⊢ 1 = if circumcenterIndex ∈ univ then Function.const (PointsWithCircumcenterIndex n) 1 circumcenterIndex else 0

Tactic:
· cases j <;> simp [circumcenterWeightsWithCircumcenter]

Goal state after:
case e'_3
n : ℕ
⊢ 1 = if circumcenterIndex ∈ univ then Function.const (PointsWithCircumcenterIndex n) 1 circumcenterIndex else 0
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‖ = 1
hy : ‖y‖ = 1
hz : ‖z‖ = 1
H : angle x z ≠ π
H0 : angle x z = angle x y + angle y z
H1 : angle x z = 0
⊢ Real.sin (angle x z) • y = Real.sin (angle y z) • x + Real.sin (angle x y) • z

Tactic:
grind [angle_nonneg, Real.sin_zero, zero_smul]

Goal state after:
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
⊢ Simplex.ExcenterExists t ∅ᶜ

pp: case inr.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
i : Fin 3
⊢ Simplex.ExcenterExists t {i}

pp: case inr.inr.inr
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
i : Fin 3
⊢ Simplex.ExcenterExists t {i}ᶜ

Tactic:
· rw [Simplex.excenterExists_compl]
    exact t.excenterExists_empty

Goal state after:
case inr.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
i : Fin 3
⊢ Simplex.ExcenterExists t {i}

case inr.inr.inr
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
i : Fin 3
⊢ Simplex.ExcenterExists t {i}ᶜ