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

Goal state before:
pp: case e'_2.e'_6
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 : Module.Oriented ℝ V (Fin 2)
inst✝ : Fact (Module.finrank ℝ V = 2)
p₁ p₂ p₃ : P
h : 2 • o.oangle (p₁ -ᵥ p₂) (p₃ -ᵥ p₂) = 2 • o.oangle (p₂ -ᵥ p₃) (p₁ -ᵥ p₃)
h0 : ∡ p₃ p₁ p₂ ≠ 0
hpi : ∡ p₃ p₁ p₂ ≠ ↑π
⊢ -(p₂ -ᵥ p₁) = p₁ -ᵥ p₂

Tactic:
simp

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

Goal state before:
pp: case e'_2.e'_7
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 : Module.Oriented ℝ V (Fin 2)
inst✝ : Fact (Module.finrank ℝ V = 2)
p₁ p₂ p₃ : P
h : 2 • o.oangle (p₁ -ᵥ p₂) (p₃ -ᵥ p₂) = 2 • o.oangle (p₂ -ᵥ p₃) (p₁ -ᵥ p₃)
h0 : ∡ p₃ p₁ p₂ ≠ 0
hpi : ∡ p₃ p₁ p₂ ≠ ↑π
⊢ -(p₂ -ᵥ p₁ - (p₃ -ᵥ p₁)) = p₃ -ᵥ p₂

Tactic:
simp

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

Goal state before:
pp: case e'_3.e'_6
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 : Module.Oriented ℝ V (Fin 2)
inst✝ : Fact (Module.finrank ℝ V = 2)
p₁ p₂ p₃ : P
h : 2 • o.oangle (p₁ -ᵥ p₂) (p₃ -ᵥ p₂) = 2 • o.oangle (p₂ -ᵥ p₃) (p₁ -ᵥ p₃)
h0 : ∡ p₃ p₁ p₂ ≠ 0
hpi : ∡ p₃ p₁ p₂ ≠ ↑π
⊢ -(p₃ -ᵥ p₁ - (p₂ -ᵥ p₁)) = p₂ -ᵥ p₃

Tactic:
simp

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

Goal state before:
pp: case e'_3.e'_7
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 : Module.Oriented ℝ V (Fin 2)
inst✝ : Fact (Module.finrank ℝ V = 2)
p₁ p₂ p₃ : P
h : 2 • o.oangle (p₁ -ᵥ p₂) (p₃ -ᵥ p₂) = 2 • o.oangle (p₂ -ᵥ p₃) (p₁ -ᵥ p₃)
h0 : ∡ p₃ p₁ p₂ ≠ 0
hpi : ∡ p₃ p₁ p₂ ≠ ↑π
⊢ -(p₃ -ᵥ p₁) = p₁ -ᵥ p₃

Tactic:
simp

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))
i : Fin (n + 1)
⊢ Set.range (s.faceOpposite i).points ⊆ Set.range s.points

Tactic:
simp

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
s₁ s₂ : Simplex ℝ P n
h : Set.range s₁.points = Set.range s₂.points
⊢ s₁.circumcenter = s₂.circumcenter

Tactic:
have hs : s₁.circumcenter ∈ affineSpan ℝ (Set.range s₂.points) :=
    h ▸ s₁.circumcenter_mem_affineSpan
  have hr : ∀ i, dist (s₂.points i) s₁.circumcenter = s₁.circumradius := by
    intro i
    have hi : s₂.points i ∈ Set.range s₂.points := Set.mem_range_self _
    rw [← h, Set.mem_range] at hi
    rcases hi with ⟨j, hj⟩
    rw [← hj, s₁.dist_circumcenter_eq_circumradius j]
  exact s₂.eq_circumcenter_of_dist_eq hs hr

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

Goal state before:
pp: case e'_3
n : ℕ
i : Fin (n + 1)
⊢ 1 = if pointIndex i ∈ univ then Function.const (PointsWithCircumcenterIndex n) 1 (pointIndex i) else 0

Tactic:
simp

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)
p : P
⊢ (s.exsphere signs).IsTangentAt p (affineSpan ℝ (Set.range (s.faceOpposite i).points)) ↔ p = s.touchpoint signs i

Tactic:
refine ⟨eq_touchpoint_of_isTangentAt_exsphere, ?_⟩
  rintro rfl
  exact h.isTangentAt_touchpoint i

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

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

Tactic:
simp

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

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
n : ℕ
inst✝ : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
hp : s.excenter signs ∈ affineSpan ℝ (Set.range s.points)
i : Fin (n + 1)
⊢ (if i ∈ signs then -1 else 1) * ↑(SignType.sign (∑ j, s.excenterWeightsUnnorm signs j)) * s.exradius signs =
    (if i ∈ signs then -1 else 1) * (↑(SignType.sign (∑ j, s.excenterWeightsUnnorm signs j)) * s.exradius signs)

Tactic:
simp

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

Goal state before:
pp: case e'_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
n : ℕ
inst✝ : NeZero n
s : Simplex ℝ P n
p : P
hp : p ∈ affineSpan ℝ (Set.range s.points)
x✝ : ℝ
a✝ : Fin (n + 1)
⊢ x✝ = (if a✝ ∈ ∅ then -1 else 1) * x✝

Tactic:
simp

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

Goal state before:
pp: case e'_5
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 j : Fin (n + 1)
hne : i ≠ j
r : ℝ
hr : r ∈ Set.Icc 0 1
hc :
  ContinuousOn
    (fun t ↦ SignType.sign ((s.signedInfDist j) ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i)) t)))
    (Set.Icc 0 1)
⊢ s.excenter signs = (AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i)) 0

Tactic:
simp

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 j : Fin (n + 1)
hne : i ≠ j
⊢ SignType.sign ((s.signedInfDist j) (s.touchpoint signs i)) =
    SignType.sign ((s.signedInfDist j) ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i)) 1))

Tactic:
simp

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

Goal state before:
pp: case mpr
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))
i : Fin (n + 1)
hw : ∑ j, s.touchpointWeights signs i j = 1
⊢ (Finset.affineCombination ℝ Finset.univ s.points) (s.touchpointWeights signs i) = s.touchpoint signs i

Tactic:
simp

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
m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
s : Simplex ℝ P n
e : Fin (n + 1) ≃ Fin (m + 1)
signs : Finset (Fin (m + 1))
i : Fin (m + 1)
⊢ (Finset.affineCombination ℝ (Finset.map e.symm.toEmbedding Finset.univ) s.points)
      (s.touchpointWeights (Finset.map e.symm.toEmbedding signs) (e.symm.toEmbedding i)) =
    (Finset.affineCombination ℝ Finset.univ s.points)
      (s.touchpointWeights (Finset.map e.symm.toEmbedding signs) (e.symm.toEmbedding i))

Tactic:
simp

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
m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
s : Simplex ℝ P n
e : Fin (n + 1) ≃ Fin (m + 1)
signs : Finset (Fin (m + 1))
i : Fin (m + 1)
⊢ (Finset.map e.symm.toEmbedding Finset.univ).sum
      (s.touchpointWeights (Finset.map e.symm.toEmbedding signs) (e.symm i)) =
    1

Tactic:
simp

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 j : Fin (n + 1)
hne : i ≠ j
hs :
  SignType.sign
      ((s.signedInfDist j) ((Finset.affineCombination ℝ Finset.univ s.points) (s.touchpointWeights signs i))) =
    SignType.sign (s.excenterWeights signs j)
⊢ ∑ i_1, s.touchpointWeights signs i i_1 = 1

Tactic:
simp

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
i j : Fin (n + 1)
hne : i ≠ j
⊢ SignType.sign (s.excenterWeights ∅ j) = 1

Tactic:
simp

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

Goal state before:
pp: case e'_4
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))
i : Fin (n + 1)
⊢ s.points '' {i}ᶜ = Set.range (s.faceOpposite i).points

Tactic:
simp

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)
⊢ ∀ (i_1 : Fin (n + 1)), ¬i_1 = i → 0 < s.touchpointWeights ∅ i i_1

Tactic:
intro j

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 j : Fin (n + 1)
⊢ ¬j = i → 0 < s.touchpointWeights ∅ i j
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)
⊢ ∀ (i_1 : Fin (n + 1)), ¬i_1 = i → 0 < s.touchpointWeights {i} i i_1

Tactic:
intro j

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 j : Fin (n + 1)
⊢ ¬j = i → 0 < s.touchpointWeights {i} i j
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.TriangleInequality

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x z : V
hxz₁ : ¬angle x z = π
hxz₂ : x ≠ 0 ∧ ∃ r, 0 < r ∧ z = r • x
left✝ : x ≠ 0
r2 : ℝ≥0
hr2 : 0 < ↑r2
hy : ↑r2 • x ≠ 0
h : angle x z = angle x (↑r2 • x) + angle (↑r2 • x) z
⊢ x ∈ {x, z}

Tactic:
simp

Goal state after:
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
y : V
hy left✝ : y ≠ 0
r : ℝ≥0
hr : 0 < ↑r
hxz₁ : ¬angle 0 (↑r • y) = π
hxz₂ : ¬angle 0 (↑r • y) = 0
h_sin_xz : Real.sin (angle 0 (↑r • y)) ≠ 0
⊢ y ∈ ℝ≥0 ∙ y

Tactic:
simp

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 : V
hx : ¬x = 0
hxz₁ : ¬angle x 0 = π
hxz₂ : ¬angle x 0 = 0
h_sin_xz : Real.sin (angle x 0) ≠ 0
left✝ : x ≠ 0
r : ℝ≥0
hr : 0 < ↑r
hy : ↑r • x ≠ 0
⊢ x ∈ {x, 0}

Tactic:
simp

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 z : V
hy : y ≠ 0
h : angle x z = angle x y + angle y z
hxz₁ : ¬angle x z = π
hxz₂ : ¬angle x z = 0
h_sin_xz : Real.sin (angle x z) ≠ 0
hx : ¬x = 0
hz : ¬z = 0
hnneg₁ : 0 ≤ Real.sin (angle y z) / ‖x‖
hnneg₂ : 0 ≤ Real.sin (angle x y) / ‖z‖
hnneg₃ : 0 ≤ Real.sin (angle x z) / ‖y‖
⊢ ⟨Real.sin (angle x z) / ‖y‖, hnneg₃⟩ ≠ 0

Tactic:
simp

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y z : V
hy : y ≠ 0
h : angle x z = angle x y + angle y z
hxz₁ : ¬angle x z = π
hxz₂ : ¬angle x z = 0
h_sin_xz : Real.sin (angle x z) ≠ 0
hx : ¬x = 0
hz : ¬z = 0
hnneg₁ : 0 ≤ Real.sin (angle y z) / ‖x‖
hnneg₂ : 0 ≤ Real.sin (angle x y) / ‖z‖
hnneg₃ : 0 ≤ Real.sin (angle x z) / ‖y‖
⊢ ¬Real.sin (angle x z) = 0 ∧ ¬y = 0
Module: Mathlib.Geometry.Euclidean.Incenter

Goal state before:
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.excenter t {i}ᶜ = Simplex.excenter t {i}

Tactic:
rw [t.excenter_compl]

Goal state after:
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
⊢ {t.points i₂, t.points i₃} = Set.range (Simplex.faceOpposite t i₁).points

Tactic:
simp

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
⊢ {t.points i₂, t.points i₃} = t.points '' {i₁}ᶜ
Module: Mathlib.Geometry.Euclidean.NinePointCircle

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)
⊢ ↑n = ‖↑n‖

Tactic:
simp

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

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
hn : NeZero 1
s : Simplex ℝ P 1
i : Fin (1 + 1)
⊢ (↑1)⁻¹ • (s.centroid -ᵥ s.points i) + (s.points i -ᵥ s.centroid) = 0

Tactic:
simp

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

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 + 2)
h : Invertible (↑n + 2 + 1)
⊢ ∑ i, centroidWeights ℝ univ i = 1

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.MetricSpace.Similarity

Goal state before:
pp: case h.«0».«1»
P₁ : Type u_3
P₂ : Type u_4
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
a b c : P₁
a' b' c' : P₂
h_ne : dist a b ≠ 0
h_ne' : dist a' b' ≠ 0
heq1 : dist a b * dist b' c' = dist b c * dist a' b'
heq2 : dist a b * dist c' a' = dist c a * dist a' b'
r : ℝ := dist a b / dist a' b'
hr : r = dist a b / dist a' b'
hr_pos : 0 < r
hij : (fun i ↦ i) ⟨0, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩
⊢ dist (![a, b, c] ((fun i ↦ i) ⟨0, ⋯⟩)) (![a, b, c] ((fun i ↦ i) ⟨1, ⋯⟩)) =
    r * dist (![a', b', c'] ((fun i ↦ i) ⟨0, ⋯⟩)) (![a', b', c'] ((fun i ↦ i) ⟨1, ⋯⟩))

Tactic:
simp

Goal state after:
case h.«0».«1»
P₁ : Type u_3
P₂ : Type u_4
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
a b c : P₁
a' b' c' : P₂
h_ne : dist a b ≠ 0
h_ne' : dist a' b' ≠ 0
heq1 : dist a b * dist b' c' = dist b c * dist a' b'
heq2 : dist a b * dist c' a' = dist c a * dist a' b'
r : ℝ := dist a b / dist a' b'
hr : r = dist a b / dist a' b'
hr_pos : 0 < r
hij : (fun i ↦ i) ⟨0, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩
⊢ dist a b = r * dist a' b'
Module: Mathlib.Topology.MetricSpace.Similarity

Goal state before:
pp: case h.«0».«2»
P₁ : Type u_3
P₂ : Type u_4
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
a b c : P₁
a' b' c' : P₂
h_ne : dist a b ≠ 0
h_ne' : dist a' b' ≠ 0
heq1 : dist a b * dist b' c' = dist b c * dist a' b'
heq2 : dist a b * dist c' a' = dist c a * dist a' b'
r : ℝ := dist a b / dist a' b'
hr : r = dist a b / dist a' b'
hr_pos : 0 < r
hij : (fun i ↦ i) ⟨0, ⋯⟩ ≠ (fun i ↦ i) ⟨2, ⋯⟩
⊢ dist (![a, b, c] ((fun i ↦ i) ⟨0, ⋯⟩)) (![a, b, c] ((fun i ↦ i) ⟨2, ⋯⟩)) =
    r * dist (![a', b', c'] ((fun i ↦ i) ⟨0, ⋯⟩)) (![a', b', c'] ((fun i ↦ i) ⟨2, ⋯⟩))

Tactic:
simp

Goal state after:
case h.«0».«2»
P₁ : Type u_3
P₂ : Type u_4
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
a b c : P₁
a' b' c' : P₂
h_ne : dist a b ≠ 0
h_ne' : dist a' b' ≠ 0
heq1 : dist a b * dist b' c' = dist b c * dist a' b'
heq2 : dist a b * dist c' a' = dist c a * dist a' b'
r : ℝ := dist a b / dist a' b'
hr : r = dist a b / dist a' b'
hr_pos : 0 < r
hij : (fun i ↦ i) ⟨0, ⋯⟩ ≠ (fun i ↦ i) ⟨2, ⋯⟩
⊢ dist a c = r * dist a' c'
Module: Mathlib.Topology.MetricSpace.Similarity

Goal state before:
pp: case h.«1».«0»
P₁ : Type u_3
P₂ : Type u_4
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
a b c : P₁
a' b' c' : P₂
h_ne : dist a b ≠ 0
h_ne' : dist a' b' ≠ 0
heq1 : dist a b * dist b' c' = dist b c * dist a' b'
heq2 : dist a b * dist c' a' = dist c a * dist a' b'
r : ℝ := dist a b / dist a' b'
hr : r = dist a b / dist a' b'
hr_pos : 0 < r
hij : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩
⊢ dist (![a, b, c] ((fun i ↦ i) ⟨1, ⋯⟩)) (![a, b, c] ((fun i ↦ i) ⟨0, ⋯⟩)) =
    r * dist (![a', b', c'] ((fun i ↦ i) ⟨1, ⋯⟩)) (![a', b', c'] ((fun i ↦ i) ⟨0, ⋯⟩))

Tactic:
simp

Goal state after:
case h.«1».«0»
P₁ : Type u_3
P₂ : Type u_4
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
a b c : P₁
a' b' c' : P₂
h_ne : dist a b ≠ 0
h_ne' : dist a' b' ≠ 0
heq1 : dist a b * dist b' c' = dist b c * dist a' b'
heq2 : dist a b * dist c' a' = dist c a * dist a' b'
r : ℝ := dist a b / dist a' b'
hr : r = dist a b / dist a' b'
hr_pos : 0 < r
hij : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩
⊢ dist b a = r * dist b' a'
Module: Mathlib.Topology.MetricSpace.Similarity

Goal state before:
pp: case h.«1».«2»
P₁ : Type u_3
P₂ : Type u_4
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
a b c : P₁
a' b' c' : P₂
h_ne : dist a b ≠ 0
h_ne' : dist a' b' ≠ 0
heq1 : dist a b * dist b' c' = dist b c * dist a' b'
heq2 : dist a b * dist c' a' = dist c a * dist a' b'
r : ℝ := dist a b / dist a' b'
hr : r = dist a b / dist a' b'
hr_pos : 0 < r
hij : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨2, ⋯⟩
⊢ dist (![a, b, c] ((fun i ↦ i) ⟨1, ⋯⟩)) (![a, b, c] ((fun i ↦ i) ⟨2, ⋯⟩)) =
    r * dist (![a', b', c'] ((fun i ↦ i) ⟨1, ⋯⟩)) (![a', b', c'] ((fun i ↦ i) ⟨2, ⋯⟩))

Tactic:
simp

Goal state after:
case h.«1».«2»
P₁ : Type u_3
P₂ : Type u_4
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
a b c : P₁
a' b' c' : P₂
h_ne : dist a b ≠ 0
h_ne' : dist a' b' ≠ 0
heq1 : dist a b * dist b' c' = dist b c * dist a' b'
heq2 : dist a b * dist c' a' = dist c a * dist a' b'
r : ℝ := dist a b / dist a' b'
hr : r = dist a b / dist a' b'
hr_pos : 0 < r
hij : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨2, ⋯⟩
⊢ dist b c = r * dist b' c'
Module: Mathlib.Topology.MetricSpace.Similarity

Goal state before:
pp: case h.«2».«0»
P₁ : Type u_3
P₂ : Type u_4
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
a b c : P₁
a' b' c' : P₂
h_ne : dist a b ≠ 0
h_ne' : dist a' b' ≠ 0
heq1 : dist a b * dist b' c' = dist b c * dist a' b'
heq2 : dist a b * dist c' a' = dist c a * dist a' b'
r : ℝ := dist a b / dist a' b'
hr : r = dist a b / dist a' b'
hr_pos : 0 < r
hij : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩
⊢ dist (![a, b, c] ((fun i ↦ i) ⟨2, ⋯⟩)) (![a, b, c] ((fun i ↦ i) ⟨0, ⋯⟩)) =
    r * dist (![a', b', c'] ((fun i ↦ i) ⟨2, ⋯⟩)) (![a', b', c'] ((fun i ↦ i) ⟨0, ⋯⟩))

Tactic:
simp

Goal state after:
case h.«2».«0»
P₁ : Type u_3
P₂ : Type u_4
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
a b c : P₁
a' b' c' : P₂
h_ne : dist a b ≠ 0
h_ne' : dist a' b' ≠ 0
heq1 : dist a b * dist b' c' = dist b c * dist a' b'
heq2 : dist a b * dist c' a' = dist c a * dist a' b'
r : ℝ := dist a b / dist a' b'
hr : r = dist a b / dist a' b'
hr_pos : 0 < r
hij : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩
⊢ dist c a = r * dist c' a'
Module: Mathlib.Topology.MetricSpace.Similarity

Goal state before:
pp: case h.«2».«1»
P₁ : Type u_3
P₂ : Type u_4
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
a b c : P₁
a' b' c' : P₂
h_ne : dist a b ≠ 0
h_ne' : dist a' b' ≠ 0
heq1 : dist a b * dist b' c' = dist b c * dist a' b'
heq2 : dist a b * dist c' a' = dist c a * dist a' b'
r : ℝ := dist a b / dist a' b'
hr : r = dist a b / dist a' b'
hr_pos : 0 < r
hij : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩
⊢ dist (![a, b, c] ((fun i ↦ i) ⟨2, ⋯⟩)) (![a, b, c] ((fun i ↦ i) ⟨1, ⋯⟩)) =
    r * dist (![a', b', c'] ((fun i ↦ i) ⟨2, ⋯⟩)) (![a', b', c'] ((fun i ↦ i) ⟨1, ⋯⟩))

Tactic:
simp

Goal state after:
case h.«2».«1»
P₁ : Type u_3
P₂ : Type u_4
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
a b c : P₁
a' b' c' : P₂
h_ne : dist a b ≠ 0
h_ne' : dist a' b' ≠ 0
heq1 : dist a b * dist b' c' = dist b c * dist a' b'
heq2 : dist a b * dist c' a' = dist c a * dist a' b'
r : ℝ := dist a b / dist a' b'
hr : r = dist a b / dist a' b'
hr_pos : 0 < r
hij : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩
⊢ dist c b = r * dist c' b'
Module: Mathlib.Geometry.Euclidean.MongePoint

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 + 2)
i₁ i₂ : Fin (n + 3)
h : ¬i₁ = i₂
⊢ ∑ i, (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i = 0

Tactic:
simp

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

Goal state before:
pp: V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b p q : P
hp : ∃ r, r • (b -ᵥ a) = p -ᵥ a
m : P := midpoint ℝ a b
hq : ‖m -ᵥ q - (m -ᵥ a)‖ = ‖m -ᵥ q + (m -ᵥ a)‖
h1 : m -ᵥ p - (m -ᵥ a) = a -ᵥ p
h1' : m -ᵥ a - (m -ᵥ p) = p -ᵥ a
h2 : m -ᵥ q - (m -ᵥ p) = p -ᵥ q
h3 : m -ᵥ q - (m -ᵥ a) = a -ᵥ q
h : ∀ (r : P), b -ᵥ r = m -ᵥ r + (m -ᵥ a)
⊢ ∃ k, m -ᵥ p = k • (m -ᵥ a)

Tactic:
rcases hp with ⟨r, hr⟩

Goal state after:
V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b p q : P
m : P := midpoint ℝ a b
hq : ‖m -ᵥ q - (m -ᵥ a)‖ = ‖m -ᵥ q + (m -ᵥ a)‖
h1 : m -ᵥ p - (m -ᵥ a) = a -ᵥ p
h1' : m -ᵥ a - (m -ᵥ p) = p -ᵥ a
h2 : m -ᵥ q - (m -ᵥ p) = p -ᵥ q
h3 : m -ᵥ q - (m -ᵥ a) = a -ᵥ q
h : ∀ (r : P), b -ᵥ r = m -ᵥ r + (m -ᵥ a)
r : ℝ
hr : r • (b -ᵥ a) = p -ᵥ a
⊢ ∃ k, m -ᵥ p = k • (m -ᵥ a)
Module: Mathlib.Geometry.Euclidean.Sphere.Power

Goal state before:
pp: V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c d p : P
h : Cospherical {a, b, c, d}
hapb : p ∈ line[ℝ, a, b]
hcpd : p ∈ line[ℝ, c, d]
q : P
r : ℝ
h' : ∀ p ∈ {a, b, c, d}, dist p q = r
⊢ a ∈ {a, b, c, d}

Tactic:
simp

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

Goal state before:
pp: V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c d p : P
h : Cospherical {a, b, c, d}
hapb : p ∈ line[ℝ, a, b]
hcpd : p ∈ line[ℝ, c, d]
q : P
r : ℝ
h' : ∀ p ∈ {a, b, c, d}, dist p q = r
⊢ b ∈ {a, b, c, d}

Tactic:
simp

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

Goal state before:
pp: V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c d p : P
h : Cospherical {a, b, c, d}
hapb : p ∈ line[ℝ, a, b]
hcpd : p ∈ line[ℝ, c, d]
q : P
r : ℝ
h' : ∀ p ∈ {a, b, c, d}, dist p q = r
⊢ c ∈ {a, b, c, d}

Tactic:
simp

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

Goal state before:
pp: V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c d p : P
h : Cospherical {a, b, c, d}
hapb : p ∈ line[ℝ, a, b]
hcpd : p ∈ line[ℝ, c, d]
q : P
r : ℝ
h' : ∀ p ∈ {a, b, c, d}, dist p q = r
⊢ d ∈ {a, b, c, d}

Tactic:
simp

Goal state after: