mathlib-initiative/mathlib-tactics huggingface dataset:
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)).SOppSide (s.points i) (s.excenter signs) ↔
    s.excenterWeights signs i < 0

Tactic:
by

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
m n : ℕ
s : Simplex ℝ P m
e : Fin (m + 1) ≃ Fin (n + 1)
⊢ (s.reindex e).circumcenter = s.circumcenter

Tactic:
by

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
m n : ℕ
s : Simplex ℝ P m
e : Fin (m + 1) ≃ Fin (n + 1)
⊢ (s.reindex e).circumradius = s.circumradius

Tactic:
by

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)
⊢ (affineSpan ℝ (Set.range (s.faceOpposite i).points)).SOppSide (s.excenter {i}) (s.points i)

Tactic:
by

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
i : Fin (n + 1)
⊢ dist ((s.map f.toAffineMap ⋯).points i) (f s.circumcenter) = s.circumradius

Tactic:
by

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)
⊢ (affineSpan ℝ (Set.range (s.faceOpposite i).points)).SOppSide (s.points i) (s.excenter {i})

Tactic:
by

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
t : Triangle ℝ P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
S : AffineSubspace ℝ P := affineSpan ℝ (Set.range t.points)
hS : S = affineSpan ℝ (Set.range t.points)
t' : Triangle ℝ ↥S := Simplex.restrict t S ⋯
⊢ finrank ℝ ↥S.direction = 2

Tactic:
by

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 j : Fin (n + 1)
h : i ≠ j
⊢ (affineSpan ℝ (Set.range (s.faceOpposite i).points)).SSameSide (s.excenter {j}) (s.points i)

Tactic:
by

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
⊢ (s.map f.toAffineMap ⋯).circumcenter = f s.circumcenter

Tactic:
by

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 j : Fin (n + 1)
h : i ≠ j
⊢ (affineSpan ℝ (Set.range (s.faceOpposite i).points)).SSameSide (s.points i) (s.excenter {j})

Tactic:
by

Goal state after:
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)
⊢ (s.map f.toAffineMap ⋯).touchpoint signs i = f (s.touchpoint signs i)

Tactic:
by

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

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
inst✝¹ : Module.Oriented ℝ V (Fin 2)
inst✝ : Fact (Module.finrank ℝ V = 2)
p₁ p₂ p₃ : P
h : 2 • ∡ p₁ p₂ p₃ = 2 • ∡ p₂ p₃ p₁
h0 : ∡ p₃ p₁ p₂ ≠ 0
hpi : ∡ p₃ p₁ p₂ ≠ ↑π
⊢ dist p₁ p₂ = dist p₁ p₃

Tactic:
by

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
S : AffineSubspace ℝ P
hS : affineSpan ℝ (Set.range s.points) ≤ S
i : Fin (n + 1)
⊢ ↑((s.restrict S hS).touchpoint signs i) = s.touchpoint signs i

Tactic:
by

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
⊢ (s.map f.toAffineMap ⋯).circumradius = s.circumradius

Tactic:
by

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)
⊢ s.touchpoint signs i ∈ affineSpan ℝ (Set.range s.points)

Tactic:
by

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ : P
h : p₂ ≠ p₁
⊢ ∠ p₁ p₂ p₃ + ∠ p₂ p₃ p₁ + ∠ p₃ p₁ p₂ = π

Tactic:
by

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
s : Simplex ℝ P n
S : AffineSubspace ℝ P
hS : affineSpan ℝ (Set.range s.points) ≤ S
⊢ ↑(s.restrict S hS).circumcenter = s.circumcenter

Tactic:
by

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ p : P
h : Sbtw ℝ p p₁ p₂
⊢ ∠ p₃ p₁ p = ∠ p₁ p₃ p₂ + ∠ p₃ p₂ p₁

Tactic:
by

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

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
inst✝¹ : Module.Oriented ℝ V (Fin 2)
inst✝ : Fact (Module.finrank ℝ V = 2)
p₁ p₂ p₃ : P
h21 : p₂ ≠ p₁
h32 : p₃ ≠ p₂
h13 : p₁ ≠ p₃
⊢ ∡ p₁ p₂ p₃ + ∡ p₂ p₃ p₁ + ∡ p₃ p₁ p₂ = ↑π

Tactic:
by

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
s : Simplex ℝ P n
S : AffineSubspace ℝ P
hS : affineSpan ℝ (Set.range s.points) ≤ S
⊢ (s.restrict S hS).circumradius = s.circumradius

Tactic:
by

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
t : Triangle ℝ P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
⊢ dist (t.points i₁) (t.points i₃) / Real.sin (∠ (t.points i₁) (t.points i₂) (t.points i₃)) / 2 = Simplex.circumradius t

Tactic:
by

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
t : Triangle ℝ P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
⊢ dist (t.points i₁) (t.points i₃) / Real.sin (∠ (t.points i₁) (t.points i₂) (t.points i₃)) = 2 * Simplex.circumradius t

Tactic:
by

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
n : ℕ
inst✝ : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i : Fin (n + 1)
⊢ (s.signedInfDist i) (s.excenter signs) =
    (if i ∈ signs then -1 else 1) * ↑(SignType.sign (∑ j, s.excenterWeightsUnnorm signs j)) * s.exradius signs

Tactic:
by

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 : Fin (n + 1)
⊢ (s.signedInfDist i) s.incenter = s.inradius

Tactic:
incenter,

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
n : ℕ
inst✝ : NeZero n
s : Simplex ℝ P n
i : Fin (n + 1)
⊢ (s.signedInfDist i) (s.exsphere ∅).center = s.inradius
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

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ p₄ : P
h : 2 • ∡ p₁ p₂ p₄ = 2 • ∡ p₁ p₃ p₄
hn : ¬Collinear ℝ {p₁, p₂, p₄}
⊢ ¬Collinear ℝ {p₁, p₃, p₄}

Tactic:
by

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c p : 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
⊢ ∠ c p b = π

Tactic:
by

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 : Fin (n + 1)
⊢ (s.signedInfDist i) s.incenter = s.inradius

Tactic:
by

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
s : Simplex ℝ P n
p : P
hr : ∃ r, ∀ (i : Fin (n + 1)), (fun x ↦ dist x p = r) (s.points i)
⊢ ∃ r, ∀ a ∈ Set.range fun i ↦ s.points i, dist a p = r

Tactic:
by

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
s : Simplex ℝ P n
p : P
hr : ∃ r, ∀ (i : Fin (n + 1)), dist (s.points i) p = r
⊢ ↑(s.orthogonalProjectionSpan p) = s.circumcenter

Tactic:
by

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
s : Simplex ℝ P n
fs : Finset (Fin (n + 1))
m : ℕ
h : #fs = m + 1
⊢ ∃ r, ∀ (i : Fin (m + 1)), dist ((s.face h).points i) s.circumcenter = r

Tactic:
by

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c p : P
hb : a ≠ b
hc : a ≠ c
hp : Wbtw ℝ b p c
⊢ ∠ b a p + ∠ p a c = ∠ b a c

Tactic:
by

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ p₄ : P
h : 2 • ∡ p₁ p₂ p₄ = 2 • ∡ p₁ p₃ p₄
hn : ¬Collinear ℝ {p₁, p₂, p₄}
hn' : ¬Collinear ℝ {p₁, p₃, p₄}
t₁ : Affine.Triangle ℝ P := { points := ![p₁, p₂, p₄], independent := ⋯ }
t₂ : Affine.Triangle ℝ P := { points := ![p₁, p₃, p₄], independent := ⋯ }
⊢ 0 ≠ 1

Tactic:
by

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a c p x : P
hx : Sbtw ℝ a x c
⊢ ∠ a p x + ∠ x p c = ∠ a p c

Tactic:
by

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)
⊢ dist (s.excenter signs) (s.touchpoint signs i) = s.exradius signs

Tactic:
by

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ p₄ : P
h : 2 • ∡ p₁ p₂ p₄ = 2 • ∡ p₁ p₃ p₄
hn : ¬Collinear ℝ {p₁, p₂, p₄}
hn' : ¬Collinear ℝ {p₁, p₃, p₄}
t₁ : Affine.Triangle ℝ P := { points := ![p₁, p₂, p₄], independent := ⋯ }
t₂ : Affine.Triangle ℝ P := { points := ![p₁, p₃, p₄], independent := ⋯ }
⊢ 0 ≠ 2

Tactic:
by

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
hs : s₁.circumcenter ∈ affineSpan ℝ (Set.range s₂.points)
⊢ ∀ (i : Fin (n + 1)), dist (s₂.points i) s₁.circumcenter = s₁.circumradius

Tactic:
by

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:
by

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ p₄ : P
h : 2 • ∡ p₁ p₂ p₄ = 2 • ∡ p₁ p₃ p₄
hn : ¬Collinear ℝ {p₁, p₂, p₄}
hn' : ¬Collinear ℝ {p₁, p₃, p₄}
t₁ : Affine.Triangle ℝ P := { points := ![p₁, p₂, p₄], independent := ⋯ }
t₂ : Affine.Triangle ℝ P := { points := ![p₁, p₃, p₄], independent := ⋯ }
⊢ 1 ≠ 2

Tactic:
by

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
x✝¹ x✝ : Fin (n + 1)
h : (fun i ↦ pointIndex i) x✝¹ = (fun i ↦ pointIndex i) x✝
⊢ x✝¹ = x✝

Tactic:
by

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
i₁ i₂ : Fin (n + 1)
⊢ dist (s.excenter signs) (s.touchpoint signs i₁) = dist (s.excenter signs) (s.touchpoint signs i₂)

Tactic:
by

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c p x : P
hx : Sbtw ℝ a x c
hxb : SameRay ℝ (x -ᵥ p) (b -ᵥ p)
hb : b ≠ p
hpx : p ≠ x
⊢ x -ᵥ p ≠ 0

Tactic:
by

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c p x : P
hx : Sbtw ℝ a x c
hxb : SameRay ℝ (x -ᵥ p) (b -ᵥ p)
hb : b ≠ p
hpx : p ≠ x
⊢ b -ᵥ p ≠ 0

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ p₄ : P
h : 2 • ∡ p₁ p₂ p₄ = 2 • ∡ p₁ p₃ p₄
hn : ¬Collinear ℝ {p₁, p₂, p₄}
⊢ Cospherical {p₁, p₂, p₃, p₄}

Tactic:
by

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c p x : P
hx : Sbtw ℝ a x c
hxb : SameRay ℝ (x -ᵥ p) (b -ᵥ p)
hb : b ≠ p
⊢ ∠ a p b + ∠ b p c = ∠ a p c

Tactic:
by

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
n : ℕ
inst✝ : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i : Fin (n + 1)
⊢ (s.exsphere signs).IsTangentAt (s.touchpoint signs i) (affineSpan ℝ (Set.range (s.faceOpposite i).points))

Tactic:
by

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

Goal state before:
pp: α : Type u_3
inst✝ : AddCommMonoid α
n : ℕ
f : PointsWithCircumcenterIndex n → α
⊢ ∑ i, f i = ∑ i, f (pointIndex i) + f circumcenterIndex

Tactic:
by

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

Tactic:
by

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

Goal state before:
pp: n : ℕ
i : Fin (n + 1)
⊢ ∑ j, pointWeightsWithCircumcenter i j = 1

Tactic:
by

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:
by

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
s : Simplex ℝ P n
i : Fin (n + 1)
⊢ pointWeightsWithCircumcenter i (pointIndex i) = 1

Tactic:
by

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c p : P
h : ∠ b p c = π
⊢ dist a b ^ 2 * dist c p + dist a c ^ 2 * dist b p = dist b c * (dist a p ^ 2 + dist b p * dist c p)

Tactic:
by

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
hf : Fact (Module.finrank ℝ V = n)
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i : Fin (n + 1)
⊢ affineSpan ℝ (Set.range (s.faceOpposite i).points) = (s.exsphere signs).orthRadius (s.touchpoint signs i)

Tactic:
by

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
s : Simplex ℝ P n
i : Fin (n + 1)
⊢ s.points i = (affineCombination ℝ univ s.pointsWithCircumcenter) (pointWeightsWithCircumcenter i)

Tactic:
by

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c : P
hbc : ¬b = c
m : P := midpoint ℝ b c
this : dist b c ≠ 0
hm :
  dist a b ^ 2 * (2⁻¹ * dist b c) + dist a c ^ 2 * (2⁻¹ * dist b c) =
    dist b c * (dist a (midpoint ℝ b c) ^ 2 + 2⁻¹ * dist b c * (2⁻¹ * dist b c))
⊢ dist a b ^ 2 + dist a c ^ 2 = 2 / dist b c * (dist a b ^ 2 * (2⁻¹ * dist b c) + dist a c ^ 2 * (2⁻¹ * dist b c))

Tactic:
by

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

Goal state before:
pp: n : ℕ
fs : Finset (Fin (n + 1))
h : fs.Nonempty
⊢ ∑ i, centroidWeightsWithCircumcenter fs i = 1

Tactic:
by

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c : P
hbc : ¬b = c
m : P := midpoint ℝ b c
this : dist b c ≠ 0
hm :
  dist a b ^ 2 * (2⁻¹ * dist b c) + dist a c ^ 2 * (2⁻¹ * dist b c) =
    dist b c * (dist a (midpoint ℝ b c) ^ 2 + 2⁻¹ * dist b c * (2⁻¹ * dist b c))
⊢ 2 / dist b c * (dist a b ^ 2 * (2⁻¹ * dist b c) + dist a c ^ 2 * (2⁻¹ * dist b c)) =
    2 * (dist a (midpoint ℝ b c) ^ 2 + (dist b c / 2) ^ 2)

Tactic:
by

Goal state after: