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

Goal state before:
pp: case pos
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
h0 : s₁.center = s₂.center
⊢ s₁.IsIntTangent s₂

pp: case neg
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
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
h0 : ¬s₁.center = s₂.center
⊢ s₁.IsIntTangent s₂

Tactic:
· rw [h0, dist_self, eq_comm, sub_eq_zero, eq_comm] at h
      have hs : s₁ = s₂ := by
        ext <;> assumption
      simp [hs, h₂]

Goal state after:
case neg
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
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
h0 : ¬s₁.center = s₂.center
⊢ s₁.IsIntTangent s₂
Module: Mathlib.Geometry.Euclidean.Circumcenter

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
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)
⊢ ∃! cs₂, cs₂.center ∈ affineSpan ℝ (insert p ↑s) ∧ insert p ps ⊆ Metric.sphere cs₂.center cs₂.radius

Tactic:
have hy0 : y ≠ 0 := dist_orthogonalProjection_ne_zero_of_notMem hp

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✝ : 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
⊢ ∃! cs₂, cs₂.center ∈ affineSpan ℝ (insert p ↑s) ∧ insert p ps ⊆ Metric.sphere cs₂.center cs₂.radius
Module: Mathlib.Geometry.Euclidean.Angle.Incenter

Goal state before:
pp: case inr.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)
h : ∡ (t.points i₂) (t.points i₁) (excenter t {i₃}) = -(∡ (excenter t {i₃}) (t.points i₁) (t.points i₂) + ↑π)
hs : excenter t signs = excenter t {i₃}
⊢ excenter t signs = incenter t ∨ excenter t signs = excenter t {i₁}

Tactic:
oangle_rev

Goal state after:
case inr.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)
h : -∡ (excenter t {i₃}) (t.points i₁) (t.points i₂) = -(∡ (excenter t {i₃}) (t.points i₁) (t.points i₂) + ↑π)
hs : excenter t signs = excenter t {i₃}
⊢ excenter t signs = incenter t ∨ excenter t signs = excenter t {i₁}
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
i : Fin (n + 1)
⊢ 0 < s.excenterWeights ∅ i

Tactic:
by
  simp only [excenterWeights, excenterWeightsUnnorm_empty_apply, Pi.smul_apply, smul_eq_mul]
  positivity

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p q : P
hp : p ∈ s
⊢ s.IsTangentAt p line[ℝ, p, q] ↔ ∠ q p s.center = π / 2

Tactic:
by
  exact ⟨fun h ↦ IsTangentAt.angle_eq_pi_div_two h (right_mem_affineSpan_pair ℝ p q),
    fun h ↦ IsTangentAt_of_angle_eq_pi_div_two h hp⟩

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

Goal state before:
pp: case refine_2.refine_2
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 j : Fin (n + 1)
hj : j ∈ {k | k ≠ i}
⊢ -(⟪s.points i -ᵥ s.altitudeFoot i, s.points j -ᵥ s.altitudeFoot j⟫ / (s.height i * s.height j)) < 1

Tactic:
rw [neg_lt]
      exact neg_one_lt_inner_vsub_altitudeFoot_div _ _ _

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)
⊢ SignType.sign (∑ i_1, s.excenterWeightsUnnorm {i} i_1)⁻¹ * SignType.sign (s.excenterWeightsUnnorm {i} i) = -1

Tactic:
convert! one_mul _

Goal state after:
case e'_2.e'_5
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 (∑ i_1, s.excenterWeightsUnnorm {i} i_1)⁻¹ = 1

case e'_3
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)
⊢ -1 = SignType.sign (s.excenterWeightsUnnorm {i} i)
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 j : Fin (n + 1)
h : i ≠ j
⊢ SignType.sign (∑ i_1, s.excenterWeightsUnnorm {i} i_1)⁻¹ * SignType.sign (s.excenterWeightsUnnorm {i} j) = 1

Tactic:
convert! one_mul _

Goal state after:
case e'_2.e'_5
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 j : Fin (n + 1)
h : i ≠ j
⊢ SignType.sign (∑ i_1, s.excenterWeightsUnnorm {i} i_1)⁻¹ = 1

case e'_3
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 j : Fin (n + 1)
h : i ≠ j
⊢ 1 = SignType.sign (s.excenterWeightsUnnorm {i} j)
Module: Mathlib.Geometry.Euclidean.Triangle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ ⟪x, x⟫ - 2 * ⟪x, y⟫ + ⟪y, y⟫ = ⟪x, x⟫ + ⟪y, y⟫ - 2 * ⟪x, y⟫

Tactic:
sub_add_eq_add_sub

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ ⟪x, x⟫ + ⟪y, y⟫ - 2 * ⟪x, y⟫ = ⟪x, x⟫ + ⟪y, y⟫ - 2 * ⟪x, y⟫
Module: Mathlib.Geometry.Euclidean.Triangle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hy : y ≠ 0
hx : x ≠ 0
h✝ : ↑(angle x y) = ↑(angle x (x + y) + angle y (x + y))
n : ℤ
this : -1 < n
h : angle y (x + y) = π
hn : 1 ≤ n
h' : angle x (x + y) = π
⊢ False

Tactic:
angle_comm,

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hy : y ≠ 0
hx : x ≠ 0
h✝ : ↑(angle x y) = ↑(angle x (x + y) + angle y (x + y))
n : ℤ
this : -1 < n
h : angle (x + y) y = π
hn : 1 ≤ n
h' : angle (x + y) x = π
⊢ False
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)
⊢ (affineSpan ℝ (Set.range (s.faceOpposite i).points)).SSameSide (s.excenter signs) (s.points i) ↔
    0 < s.excenterWeights signs i

Tactic:
by
  rw [excenter_eq_affineCombination,
    s.sSameSide_affineSpan_faceOpposite_point_right_iff h.sum_excenterWeights_eq_one]

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MetricSpace P
inst✝⁴ : NormedAddTorsor V P
V₂ : Type u_3
P₂ : Type u_4
inst✝³ : NormedAddCommGroup V₂
inst✝² : InnerProductSpace ℝ V₂
inst✝¹ : MetricSpace P₂
inst✝ : NormedAddTorsor V₂ P₂
n : ℕ
s : Simplex ℝ P n
f : P →ᵃⁱ[ℝ] P₂
⊢ f s.circumcenter ∈ affineSpan ℝ (Set.range (s.map f.toAffineMap ⋯).points)

Tactic:
map_points,

Goal state after:
V : Type u_1
P : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MetricSpace P
inst✝⁴ : NormedAddTorsor V P
V₂ : Type u_3
P₂ : Type u_4
inst✝³ : NormedAddCommGroup V₂
inst✝² : InnerProductSpace ℝ V₂
inst✝¹ : MetricSpace P₂
inst✝ : NormedAddTorsor V₂ P₂
n : ℕ
s : Simplex ℝ P n
f : P →ᵃⁱ[ℝ] P₂
⊢ f s.circumcenter ∈ affineSpan ℝ (Set.range (⇑f.toAffineMap ∘ s.points))
Module: Mathlib.Geometry.Euclidean.Triangle

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
a b c p : P
hb : a ≠ b
hc : a ≠ c
pb : ¬p = b
pc : ¬p = c
ea : ∠ b a c + ∠ a c b + ∠ c b a = π
eb : ∠ b a p + ∠ a p b + ∠ p b a = π
ec : ∠ a c p + ∠ c p a + ∠ p a c = π
hp : ∠ b p c = π
hp' : ∠ c p b = π
⊢ ∠ b a p + ∠ p a c = ∠ b a c

Tactic:
angle_comm p b a,

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
a b c p : P
hb : a ≠ b
hc : a ≠ c
pb : ¬p = b
pc : ¬p = c
ea : ∠ b a c + ∠ a c b + ∠ c b a = π
eb : ∠ b a p + ∠ a p b + ∠ a b p = π
ec : ∠ a c p + ∠ c p a + ∠ p a c = π
hp : ∠ b p c = π
hp' : ∠ c p b = π
⊢ ∠ b a p + ∠ p a c = ∠ b a c
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
s : Simplex ℝ P n
p : P
hr✝ : ∃ r, ∀ (i : Fin (n + 1)), (fun x ↦ dist x p = r) (s.points i)
hr : ∃ r, ∀ a ∈ Set.range fun i ↦ s.points i, dist a p = r
⊢ ↑(s.orthogonalProjectionSpan p) = s.circumcenter

Tactic:
rw [exists_dist_eq_iff_exists_dist_orthogonalProjection_eq (subset_affineSpan ℝ _) p] at 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
n : ℕ
s : Simplex ℝ P n
p : P
hr✝ : ∃ r, ∀ (i : Fin (n + 1)), (fun x ↦ dist x p = r) (s.points i)
hr :
  ∃ r,
    ∀ p₁ ∈ Set.range fun i ↦ s.points i,
      dist p₁ ↑((orthogonalProjection (affineSpan ℝ (Set.range fun i ↦ s.points i))) p) = r
⊢ ↑(s.orthogonalProjectionSpan p) = s.circumcenter
Module: Mathlib.Geometry.Euclidean.Triangle

Goal state before:
pp: case 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
a b c p : P
hb : b ≠ p
hx : Sbtw ℝ a p c
hxb : SameRay ℝ (p -ᵥ p) (b -ᵥ p)
hpi : ∠ a p c = π
⊢ ∠ a p b + ∠ b p c = ∠ a p c

Tactic:
rw [hpi, angle_comm a p b]

Goal state after:
case 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
a b c p : P
hb : b ≠ p
hx : Sbtw ℝ a p c
hxb : SameRay ℝ (p -ᵥ p) (b -ᵥ p)
hpi : ∠ a p c = π
⊢ ∠ b p a + ∠ b p c = π
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 : P
h : ∠ b p c = π
⊢ (dist a p * dist a p + dist b p * dist b p - 2 * dist a p * dist b p * -Real.cos (∠ a p c)) * dist c p +
      (dist a p * dist a p + dist c p * dist c p - 2 * dist a p * dist c p * Real.cos (∠ a p c)) * dist b p =
    dist b c * (dist a p ^ 2 + dist b p * dist c p)

Tactic:
dist_eq_add_dist_of_angle_eq_pi h

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
a b c p : P
h : ∠ b p c = π
⊢ (dist a p * dist a p + dist b p * dist b p - 2 * dist a p * dist b p * -Real.cos (∠ a p c)) * dist c p +
      (dist a p * dist a p + dist c p * dist c p - 2 * dist a p * dist c p * Real.cos (∠ a p c)) * dist b p =
    (dist b p + dist c p) * (dist a p ^ 2 + dist b p * dist c p)
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
s : Simplex ℝ P n
i₁ i₂ : Fin (n + 1)
h : i₁ ≠ i₂
hc : #{i₁, i₂} = 2
W : AffineSubspace ℝ P := affineSpan ℝ (s.points '' {i₁, i₂})
h_faces : ↑((orthogonalProjection W) s.circumcenter) = ↑((s.face hc).orthogonalProjectionSpan s.circumcenter)
⊢ ∑ x,
        ((centroidWeightsWithCircumcenter {i₁, i₂} - circumcenterWeightsWithCircumcenter n +
                centroidWeightsWithCircumcenter {i₁, i₂})
              (pointIndex x) -
            reflectionCircumcenterWeightsWithCircumcenter i₁ i₂ (pointIndex x)) •
          (s.pointsWithCircumcenter (pointIndex x) -ᵥ Classical.choice ⋯) +
      ((centroidWeightsWithCircumcenter {i₁, i₂} - circumcenterWeightsWithCircumcenter n +
              centroidWeightsWithCircumcenter {i₁, i₂})
            circumcenterIndex -
          reflectionCircumcenterWeightsWithCircumcenter i₁ i₂ circumcenterIndex) •
        (s.pointsWithCircumcenter circumcenterIndex -ᵥ Classical.choice ⋯) =
    0

Tactic:
Pi.add_apply,

Goal state after:
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P n
i₁ i₂ : Fin (n + 1)
h : i₁ ≠ i₂
hc : #{i₁, i₂} = 2
W : AffineSubspace ℝ P := affineSpan ℝ (s.points '' {i₁, i₂})
h_faces : ↑((orthogonalProjection W) s.circumcenter) = ↑((s.face hc).orthogonalProjectionSpan s.circumcenter)
⊢ ∑ x,
        ((centroidWeightsWithCircumcenter {i₁, i₂} - circumcenterWeightsWithCircumcenter n) (pointIndex x) +
              centroidWeightsWithCircumcenter {i₁, i₂} (pointIndex x) -
            reflectionCircumcenterWeightsWithCircumcenter i₁ i₂ (pointIndex x)) •
          (s.pointsWithCircumcenter (pointIndex x) -ᵥ Classical.choice ⋯) +
      ((centroidWeightsWithCircumcenter {i₁, i₂} - circumcenterWeightsWithCircumcenter n) circumcenterIndex +
            centroidWeightsWithCircumcenter {i₁, i₂} circumcenterIndex -
          reflectionCircumcenterWeightsWithCircumcenter i₁ i₂ circumcenterIndex) •
        (s.pointsWithCircumcenter circumcenterIndex -ᵥ Classical.choice ⋯) =
    0
Module: Mathlib.Geometry.Euclidean.Circumcenter

Goal state before:
pp: case mp
V : Type u_1
P : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
s : AffineSubspace ℝ P
ps : Set P
h : ps ⊆ ↑s
inst✝¹ : Nonempty ↥s
inst✝ : s.direction.HasOrthogonalProjection
c : P
hcr : ∃ r, ∀ p₁ ∈ ps, dist p₁ ↑((orthogonalProjection s) c) = r
⊢ ∃ center ∈ s, ∃ radius, ∀ p ∈ ps, dist p center = radius

Tactic:
exact ⟨orthogonalProjection s c, orthogonalProjection_mem _, hcr⟩

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

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : ‖x‖ = 1
hy : ‖y‖ = 1
hxy : angle x y = 0
⊢ x = y

Tactic:
by
  grind [eq_of_angle_eq_zero_of_norm_eq]

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
ps : Set P
n : ℕ
inst✝ : FiniteDimensional ℝ V
hd : finrank ℝ V = n
hc : Cospherical ps
sx₁ sx₂ : Simplex ℝ P n
hsx₁ : Set.range sx₁.points ⊆ ps
hsx₂ : Set.range sx₂.points ⊆ ps
⊢ sx₁.circumcenter = sx₂.circumcenter

Tactic:
rcases exists_circumcenter_eq_of_cospherical hd hc with ⟨r, hr⟩

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
ps : Set P
n : ℕ
inst✝ : FiniteDimensional ℝ V
hd : finrank ℝ V = n
hc : Cospherical ps
sx₁ sx₂ : Simplex ℝ P n
hsx₁ : Set.range sx₁.points ⊆ ps
hsx₂ : Set.range sx₂.points ⊆ ps
r : P
hr : ∀ (sx : Simplex ℝ P n), Set.range sx.points ⊆ ps → sx.circumcenter = r
⊢ sx₁.circumcenter = sx₂.circumcenter
Module: Mathlib.Geometry.Euclidean.Incenter

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁸ : NormedAddCommGroup V
inst✝⁷ : InnerProductSpace ℝ V
inst✝⁶ : MetricSpace P
inst✝⁵ : NormedAddTorsor V P
V₂ : Type u_3
P₂ : Type u_4
inst✝⁴ : NormedAddCommGroup V₂
inst✝³ : InnerProductSpace ℝ V₂
inst✝² : MetricSpace P₂
inst✝¹ : NormedAddTorsor V₂ P₂
n : ℕ
inst✝ : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
f : P →ᵃⁱ[ℝ] P₂
i : Fin (n + 1)
hc :
  (Finset.affineCombination ℝ Finset.univ (s.map f.toAffineMap ⋯).points)
      ((s.map f.toAffineMap ⋯).touchpointWeights signs i) =
    f (s.touchpoint signs i)
⊢ (Finset.affineCombination ℝ Finset.univ s.points) ((s.map f.toAffineMap ⋯).touchpointWeights signs i) =
    s.touchpoint signs i

Tactic:
map_points,

Goal state after:
V : Type u_1
P : Type u_2
inst✝⁸ : NormedAddCommGroup V
inst✝⁷ : InnerProductSpace ℝ V
inst✝⁶ : MetricSpace P
inst✝⁵ : NormedAddTorsor V P
V₂ : Type u_3
P₂ : Type u_4
inst✝⁴ : NormedAddCommGroup V₂
inst✝³ : InnerProductSpace ℝ V₂
inst✝² : MetricSpace P₂
inst✝¹ : NormedAddTorsor V₂ P₂
n : ℕ
inst✝ : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
f : P →ᵃⁱ[ℝ] P₂
i : Fin (n + 1)
hc :
  (Finset.affineCombination ℝ Finset.univ (⇑f.toAffineMap ∘ s.points))
      ((s.map f.toAffineMap ⋯).touchpointWeights signs i) =
    f (s.touchpoint signs i)
⊢ (Finset.affineCombination ℝ Finset.univ s.points) ((s.map f.toAffineMap ⋯).touchpointWeights signs i) =
    s.touchpoint signs i
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.TriangleInequality

Goal state before:
pp: case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x z : V
kx kz : ℝ≥0
hy : kx • x + kz • z ≠ 0
hkx : 0 < kx
hkz : 0 < kz
hz : z = 0
hx : x ≠ 0
⊢ angle x z = angle x (kx • x + kz • z) + angle z (kx • x + kz • z)

Tactic:
angle_comm,

Goal state after:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x z : V
kx kz : ℝ≥0
hy : kx • x + kz • z ≠ 0
hkx : 0 < kx
hkz : 0 < kz
hz : z = 0
hx : x ≠ 0
⊢ angle z x = angle x (kx • x + kz • z) + angle z (kx • x + kz • z)
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.TriangleInequality

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y z : V
hy : y ≠ 0
h_mem : y ∈ Submodule.span ℝ≥0 {x, z}
⊢ angle x z = angle x y + angle y z

Tactic:
rw [Submodule.mem_span_pair] at h_mem
  obtain ⟨kx, kz, rfl⟩ := h_mem
  rcases eq_zero_or_pos kx with rfl | hkx <;> rcases eq_zero_or_pos kz with rfl | hkz
  · simp at hy
  · simp_all [NNReal.smul_def]
  · simp_all [NNReal.smul_def]
  · rw [angle_comm (_ + _) z]
    by_cases! hz : z ≠ 0
    · simpa [hkx, hkz, NNReal.smul_def] using
        angle_eq_angle_add_add_angle_add (kx • x) (smul_ne_zero hkz.ne' hz)
    · have hx : x ≠ 0 := by simp_all
      rw [angle_comm, add_comm, add_comm (kx • x)]
      simpa [hkx, hkz, NNReal.smul_def] using
        angle_eq_angle_add_add_angle_add (kz • z) (smul_ne_zero hkx.ne' hx)

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

Goal state before:
pp: V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst✝⁷ : NormedAddCommGroup V₁
inst✝⁶ : NormedAddCommGroup V₂
inst✝⁵ : InnerProductSpace ℝ V₁
inst✝⁴ : InnerProductSpace ℝ V₂
inst✝³ : MetricSpace P₁
inst✝² : MetricSpace P₂
inst✝¹ : NormedAddTorsor V₁ P₁
inst✝ : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h : ¬Collinear ℝ {a, b, c}
ha₁ : ∠ a b c = ∠ a' b' c'
hd : dist b c = dist b' c'
ha₂ : ∠ b c a = ∠ b' c' a'
h' : ¬Collinear ℝ {a', b', c'}
ha₃' : ∠ c' a' b' + ∠ a' b' c' + ∠ b' c' a' = Real.pi
ha₃ : ∠ c a b = ∠ c' a' b'
h_bac : ¬Collinear ℝ {b, a, c}
h_bac' : ¬Collinear ℝ {b', a', c'}
⊢ dist b' c' * Real.sin (∠ a c b) / Real.sin (∠ b a c) = dist b' c' * Real.sin (∠ a' c' b') / Real.sin (∠ b' a' c')

Tactic:
angle_comm,

Goal state after:
V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst✝⁷ : NormedAddCommGroup V₁
inst✝⁶ : NormedAddCommGroup V₂
inst✝⁵ : InnerProductSpace ℝ V₁
inst✝⁴ : InnerProductSpace ℝ V₂
inst✝³ : MetricSpace P₁
inst✝² : MetricSpace P₂
inst✝¹ : NormedAddTorsor V₁ P₁
inst✝ : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h : ¬Collinear ℝ {a, b, c}
ha₁ : ∠ a b c = ∠ a' b' c'
hd : dist b c = dist b' c'
ha₂ : ∠ b c a = ∠ b' c' a'
h' : ¬Collinear ℝ {a', b', c'}
ha₃' : ∠ c' a' b' + ∠ a' b' c' + ∠ b' c' a' = Real.pi
ha₃ : ∠ c a b = ∠ c' a' b'
h_bac : ¬Collinear ℝ {b, a, c}
h_bac' : ¬Collinear ℝ {b', a', c'}
⊢ dist b' c' * Real.sin (∠ b c a) / Real.sin (∠ b a c) = dist b' c' * Real.sin (∠ a' c' b') / Real.sin (∠ b' a' c')
Module: Mathlib.Geometry.Euclidean.MongePoint

Goal state before:
pp: case e_6
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 2)
i : PointsWithCircumcenterIndex (n + 2)
⊢ ((↑(n + 2 + 1) / ↑(n + 2 - 1)) •
          (centroidWeightsWithCircumcenter univ - circumcenterWeightsWithCircumcenter (n + 2)) +
        circumcenterWeightsWithCircumcenter (n + 2))
      i =
    mongePointWeightsWithCircumcenter n i

Tactic:
Pi.add_apply,

Goal state after:
case e_6
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 2)
i : PointsWithCircumcenterIndex (n + 2)
⊢ ((↑(n + 2 + 1) / ↑(n + 2 - 1)) • (centroidWeightsWithCircumcenter univ - circumcenterWeightsWithCircumcenter (n + 2)))
        i +
      circumcenterWeightsWithCircumcenter (n + 2) i =
    mongePointWeightsWithCircumcenter n i
Module: Mathlib.Geometry.Euclidean.Simplex

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P n
he : s.Equilateral
i₁ i₂ i₃ : Fin (n + 1)
⊢ i₁ ≠ i₂ → i₁ ≠ i₃ → i₂ ≠ i₃ → ∠ (s.points i₁) (s.points i₂) (s.points i₃) < π / 2

Tactic:
h₁₂

Goal state after:
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P n
he : s.Equilateral
i₁ i₂ i₃ : Fin (n + 1)
h₁₂ : i₁ ≠ i₂
⊢ i₁ ≠ i₃ → i₂ ≠ i₃ → ∠ (s.points i₁) (s.points i₂) (s.points i₃) < π / 2