mathlib-initiative/mathlib-tactics huggingface dataset:
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 x y = ↑(π / 2)
⊢ (o.oangle x (x + y)).tan = ‖y‖ / ‖x‖

Tactic:
have hs : (o.oangle x (x + y)).sign = 1 := by
    rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]

Goal state after:
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 x y = ↑(π / 2)
hs : (o.oangle x (x + y)).sign = 1
⊢ (o.oangle x (x + y)).tan = ‖y‖ / ‖x‖
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 x y = ↑(π / 2)
⊢ (o.oangle x (x + y)).cos * ‖x + y‖ = ‖x‖

Tactic:
have hs : (o.oangle x (x + y)).sign = 1 := by
    rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]

Goal state after:
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 x y = ↑(π / 2)
hs : (o.oangle x (x + y)).sign = 1
⊢ (o.oangle x (x + y)).cos * ‖x + y‖ = ‖x‖
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 x y = ↑(π / 2)
⊢ (o.oangle x (x + y)).sin * ‖x + y‖ = ‖y‖

Tactic:
have hs : (o.oangle x (x + y)).sign = 1 := by
    rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]

Goal state after:
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 x y = ↑(π / 2)
hs : (o.oangle x (x + y)).sign = 1
⊢ (o.oangle x (x + y)).sin * ‖x + y‖ = ‖y‖
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 x y = ↑(π / 2)
⊢ (o.oangle x (x + y)).tan * ‖x‖ = ‖y‖

Tactic:
have hs : (o.oangle x (x + y)).sign = 1 := by
    rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]

Goal state after:
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 x y = ↑(π / 2)
hs : (o.oangle x (x + y)).sign = 1
⊢ (o.oangle x (x + y)).tan * ‖x‖ = ‖y‖
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 x y = ↑(π / 2)
⊢ ‖x‖ / (o.oangle x (x + y)).cos = ‖x + y‖

Tactic:
have hs : (o.oangle x (x + y)).sign = 1 := by
    rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]

Goal state after:
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 x y = ↑(π / 2)
hs : (o.oangle x (x + y)).sign = 1
⊢ ‖x‖ / (o.oangle x (x + y)).cos = ‖x + y‖
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 x y = ↑(π / 2)
⊢ ‖y‖ / (o.oangle x (x + y)).sin = ‖x + y‖

Tactic:
have hs : (o.oangle x (x + y)).sign = 1 := by
    rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]

Goal state after:
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 x y = ↑(π / 2)
hs : (o.oangle x (x + y)).sign = 1
⊢ ‖y‖ / (o.oangle x (x + y)).sin = ‖x + y‖
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 x y = ↑(π / 2)
⊢ ‖y‖ / (o.oangle x (x + y)).tan = ‖x‖

Tactic:
have hs : (o.oangle x (x + y)).sign = 1 := by
    rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]

Goal state after:
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 x y = ↑(π / 2)
hs : (o.oangle x (x + y)).sign = 1
⊢ ‖y‖ / (o.oangle x (x + y)).tan = ‖x‖
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Affine

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
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₁' p₂ p₃ : P
h : Wbtw ℝ p₂ p₁ p₁'
hp₁p₂ : p₁ ≠ p₂
hp₃p₂ : ¬p₃ = p₂
hp₁'p₂ : p₁' = p₂
⊢ ∡ p₁ p₂ p₃ = ∡ p₁' p₂ p₃

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
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₁' p₂ p₃ : P
h : Wbtw ℝ p₂ p₁ p₁'
hp₁p₂ : p₁ ≠ p₂
hp₃p₂ : ¬p₃ = p₂
hp₁'p₂ : ¬p₁' = p₂
⊢ ∡ p₁ p₂ p₃ = ∡ p₁' p₂ p₃

Tactic:
· rw [hp₁'p₂, wbtw_self_iff] at h; exact False.elim (hp₁p₂ h)

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
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₁' p₂ p₃ : P
h : Wbtw ℝ p₂ p₁ p₁'
hp₁p₂ : p₁ ≠ p₂
hp₃p₂ : ¬p₃ = p₂
hp₁'p₂ : ¬p₁' = p₂
⊢ ∡ p₁ p₂ p₃ = ∡ p₁' p₂ p₃
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

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 : ∠ p₁ p₂ p₃ = π / 2
h0 : p₁ ≠ p₂ ∨ p₃ = p₂
⊢ dist p₁ p₂ / Real.tan (∠ p₂ p₃ p₁) = dist p₃ p₂

Tactic:
by
  rw [angle, ← inner_eq_zero_iff_angle_eq_pi_div_two, real_inner_comm, ← neg_eq_zero, ←
    inner_neg_left, neg_vsub_eq_vsub_rev] at h
  rw [eq_comm, ← @vsub_ne_zero V, ← @vsub_eq_zero_iff_eq V, or_comm] at h0
  rw [angle, dist_eq_norm_vsub V p₁ p₂, dist_eq_norm_vsub' V p₃ p₂, ← vsub_add_vsub_cancel p₁ p₂ p₃,
    add_comm, norm_div_tan_angle_add_of_inner_eq_zero h h0]

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

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
v : V
p q r : P
⊢ ⟪NormedSpace.normalize v, r -ᵥ q⟫ + ⟪NormedSpace.normalize v, q -ᵥ p⟫ = ⟪NormedSpace.normalize v, r -ᵥ p⟫

Tactic:
← inner_add_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
v : V
p q r : P
⊢ ⟪NormedSpace.normalize v, r -ᵥ q + (q -ᵥ p)⟫ = ⟪NormedSpace.normalize v, r -ᵥ p⟫
Module: Mathlib.Geometry.Euclidean.SignedDist

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
v : V
p q : P
h : ⟪v, p -ᵥ q⟫ = 0
r : P
⊢ ((signedDist v) p) r = ((signedDist v) q) r

Tactic:
simpa [NormedSpace.normalize, real_inner_smul_left, h] using signedDist_vadd_left v (p -ᵥ q) q r

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
p₁ p₂ : P
⊢ (perpBisector p₁ p₂).direction = (ℝ ∙ (p₂ -ᵥ p₁))ᗮ

Tactic:
rw [perpBisector, direction_mk']

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
⊢ ((innerₛₗ ℝ) (p₂ -ᵥ p₁)).ker = (ℝ ∙ (p₂ -ᵥ p₁))ᗮ
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
r : ℝ
v : V
p₁ p₂ : P
⊢ ⟪r • v + (p₁ -ᵥ p₂), r • v + (p₁ -ᵥ p₂)⟫ = ⟪v, v⟫ * r * r + 2 * ⟪v, p₁ -ᵥ p₂⟫ * r + ⟪p₁ -ᵥ p₂, p₁ -ᵥ p₂⟫

Tactic:
real_inner_add_add_self,

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
r : ℝ
v : V
p₁ p₂ : P
⊢ ⟪r • v, r • v⟫ + 2 * ⟪r • v, p₁ -ᵥ p₂⟫ + ⟪p₁ -ᵥ p₂, p₁ -ᵥ p₂⟫ =
    ⟪v, v⟫ * r * r + 2 * ⟪v, p₁ -ᵥ p₂⟫ * r + ⟪p₁ -ᵥ p₂, p₁ -ᵥ p₂⟫
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Affine

Goal state before:
pp: case refine_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
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ p₄ : P
hp₁ : p₁ ≠ p₂
hp₄ : p₄ ≠ p₂
h : SameRay ℝ (p₁ -ᵥ p₂) (p₄ -ᵥ p₂)
⊢ Wbtw ℝ p₂ p₁ p₄ ∨ Wbtw ℝ p₂ p₄ p₁

Tactic:
obtain ⟨r, hr, he⟩ := h.exists_pos_left (vsub_ne_zero.2 hp₁) (vsub_ne_zero.2 hp₄)

Goal state after:
case refine_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
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ p₄ : P
hp₁ : p₁ ≠ p₂
hp₄ : p₄ ≠ p₂
h : SameRay ℝ (p₁ -ᵥ p₂) (p₄ -ᵥ p₂)
r : ℝ
hr : 0 < r
he : r • (p₁ -ᵥ p₂) = p₄ -ᵥ p₂
⊢ Wbtw ℝ p₂ p₁ p₄ ∨ Wbtw ℝ p₂ p₄ p₁
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
hadd : ∡ p₁ p₂ p₃ + ∡ p₃ p₂ p₄ = ∡ p₁ p₂ p₄
h : ∡ p₁ p₂ p₃ ≠ ↑π
⊢ (∡ p₁ p₂ p₃).sign = (∡ p₁ p₂ p₃ + ∡ p₃ p₂ p₄).sign

Tactic:
hadd,

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
hadd : ∡ p₁ p₂ p₃ + ∡ p₃ p₂ p₄ = ∡ p₁ p₂ p₄
h : ∡ p₁ p₂ p₃ ≠ ↑π
⊢ (∡ p₁ p₂ p₃).sign = (∡ p₁ p₂ p₄).sign
Module: Mathlib.Geometry.Euclidean.Angle.Bisector

Goal state before:
pp: case inl
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 p₂ p₃ : P
h₂ : p ≠ p₂
h : 2 • ∡ p₂ p p = 2 • ∡ p p p₃
h' : ↑((orthogonalProjection line[ℝ, p, p₂]) p) = p
⊢ dist p ↑((orthogonalProjection line[ℝ, p, p₂]) p) = dist p ↑((orthogonalProjection line[ℝ, p, p₃]) p)

Tactic:
h',

Goal state after:
case inl
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 p₂ p₃ : P
h₂ : p ≠ p₂
h : 2 • ∡ p₂ p p = 2 • ∡ p p p₃
h' : ↑((orthogonalProjection line[ℝ, p, p₂]) p) = p
⊢ dist p p = dist p ↑((orthogonalProjection line[ℝ, p, p₃]) p)
Module: Mathlib.Geometry.Euclidean.Angle.Bisector

Goal state before:
pp: case e_a
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' : ↑((orthogonalProjection line[ℝ, p₁, p₂]) p) = p₁
hp : p ≠ ↑((orthogonalProjection line[ℝ, p₁, p₂]) p)
hpm : p ∉ line[ℝ, p₁, p₂]
h : inner ℝ (p -ᵥ p₁) (p₃ -ᵥ p₁) = 0
⊢ ↑((orthogonalProjection line[ℝ, p₁, p₂]) p) = ↑((orthogonalProjection line[ℝ, p₁, p₃]) p)

Tactic:
h',

Goal state after:
case e_a
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' : ↑((orthogonalProjection line[ℝ, p₁, p₂]) p) = p₁
hp : p ≠ ↑((orthogonalProjection line[ℝ, p₁, p₂]) p)
hpm : p ∉ line[ℝ, p₁, p₂]
h : inner ℝ (p -ᵥ p₁) (p₃ -ᵥ p₁) = 0
⊢ p₁ = ↑((orthogonalProjection line[ℝ, p₁, p₃]) p)
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 q : P
⊢ ‖q -ᵥ s.center‖ * ‖q -ᵥ s.center‖ = ‖p -ᵥ s.center‖ * ‖p -ᵥ s.center‖ + ‖q -ᵥ p‖ * ‖q -ᵥ p‖ ↔ q ∈ s.orthRadius p

Tactic:
← vsub_add_vsub_cancel q p s.center

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 q : P
⊢ ‖q -ᵥ p + (p -ᵥ s.center)‖ * ‖q -ᵥ p + (p -ᵥ s.center)‖ = ‖p -ᵥ s.center‖ * ‖p -ᵥ s.center‖ + ‖q -ᵥ p‖ * ‖q -ᵥ p‖ ↔
    q ∈ s.orthRadius p
Module: Mathlib.Geometry.Euclidean.Sphere.Tangent

Goal state before:
pp: case refine_1
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
as : AffineSubspace ℝ P
p : P
h : s.IsTangentAt p as
⊢ Metric.infDist s.center ↑as ≤ s.radius

Tactic:
convert! Metric.infDist_le_dist_of_mem h.mem_space
    rw [mem_sphere'.1 h.mem_sphere]

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

Goal state before:
pp: case neg.refine_2
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
p₁ p₂ : P
hp₁ : dist p₁ s.center = s.radius
hp₂ : dist p₂ s.center ≤ s.radius
h : ¬p₁ = p₂
⊢ ⟪p₂ -ᵥ s.center, p₁ -ᵥ s.center⟫ ≠ ‖p₁ -ᵥ s.center‖ * ‖p₁ -ᵥ s.center‖

Tactic:
rcases hp₂.lt_or_eq with (hp₂' | hp₂')
    · refine ((real_inner_le_norm _ _).trans_lt (mul_lt_mul_of_pos_right ?_ ?_)).ne
      · rwa [← hp₁, @dist_eq_norm_vsub V, @dist_eq_norm_vsub V] at hp₂'
      · rw [norm_pos_iff, vsub_ne_zero]
        rintro rfl
        rw [← hp₁] at hp₂'
        refine (dist_nonneg.not_gt : ¬dist p₂ s.center < 0) ?_
        simpa using hp₂'
    · rw [← hp₁, @dist_eq_norm_vsub V, @dist_eq_norm_vsub V] at hp₂'
      nth_rw 1 [← hp₂']
      rw [Ne, inner_eq_norm_mul_iff_real, hp₂', ← sub_eq_zero, ← smul_sub,
        vsub_sub_vsub_cancel_right, ← Ne, smul_ne_zero_iff, vsub_ne_zero,
        and_iff_left (Ne.symm h), norm_ne_zero_iff, vsub_ne_zero]
      rintro rfl
      refine h (Eq.symm ?_)
      simpa using hp₂'

Goal state after:
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
p : P
ha : AffineIndependent ℝ ![t.points i₁, t.points i₂, t.points i₃]
⊢ dist p ↑((orthogonalProjection (affineSpan ℝ (t.points '' {i₁, i₂}))) p) =
      dist p ↑((orthogonalProjection (affineSpan ℝ (t.points '' {i₁, i₃}))) p) ↔
    dist p ↑((orthogonalProjection line[ℝ, t.points i₁, t.points i₂]) p) =
      dist p ↑((orthogonalProjection line[ℝ, t.points i₁, t.points i₃]) p)

Tactic:
Set.image_insert_eq,

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
p : P
ha : AffineIndependent ℝ ![t.points i₁, t.points i₂, t.points i₃]
⊢ dist p ↑((orthogonalProjection (affineSpan ℝ (insert (t.points i₁) (t.points '' {i₂})))) p) =
      dist p ↑((orthogonalProjection (affineSpan ℝ (insert (t.points i₁) (t.points '' {i₃})))) p) ↔
    dist p ↑((orthogonalProjection line[ℝ, t.points i₁, t.points i₂]) p) =
      dist p ↑((orthogonalProjection line[ℝ, t.points i₁, t.points i₃]) p)
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
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
ycc₂ : ℝ := (x * x + y * y - cr * cr) / (2 * y)
cc₂ : P := (ycc₂ / y) • (p -ᵥ ↑((orthogonalProjection s) p)) +ᵥ cc
cr₂ : ℝ := √(cr * cr + ycc₂ * ycc₂)
hpo : p = 1 • (p -ᵥ ↑((orthogonalProjection s) p)) +ᵥ ↑((orthogonalProjection s) p)
cc₃ : P
cr₃ : ℝ
hcc₃ : cc₃ ∈ affineSpan ℝ (insert p ↑s)
hcr₃ : insert p ps ⊆ Metric.sphere cc₃ cr₃
t₃ : ℝ
hcc₃' : cc ∈ s
hcc₃'' : cc₃ = t₃ • (p -ᵥ ↑((orthogonalProjection s) p)) +ᵥ cc
hcr₃' : ∀ p₁ ∈ ps, dist p₁ ↑⟨cc, hcc₃'⟩ = cr
p0 : P
hp0 : p0 ∈ ps
h' : ↑⟨cc, hcc₃'⟩ = cc
⊢ dist p0 ↑⟨cc, hcc₃'⟩ * dist p0 ↑⟨cc, hcc₃'⟩ +
      dist (t₃ • (p -ᵥ ↑((orthogonalProjection s) p)) +ᵥ cc) ↑⟨cc, hcc₃'⟩ *
        dist (t₃ • (p -ᵥ ↑((orthogonalProjection s) p)) +ᵥ cc) ↑⟨cc, hcc₃'⟩ =
    cr * cr + t₃ * y * (t₃ * y)

Tactic:
h',

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
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
ycc₂ : ℝ := (x * x + y * y - cr * cr) / (2 * y)
cc₂ : P := (ycc₂ / y) • (p -ᵥ ↑((orthogonalProjection s) p)) +ᵥ cc
cr₂ : ℝ := √(cr * cr + ycc₂ * ycc₂)
hpo : p = 1 • (p -ᵥ ↑((orthogonalProjection s) p)) +ᵥ ↑((orthogonalProjection s) p)
cc₃ : P
cr₃ : ℝ
hcc₃ : cc₃ ∈ affineSpan ℝ (insert p ↑s)
hcr₃ : insert p ps ⊆ Metric.sphere cc₃ cr₃
t₃ : ℝ
hcc₃' : cc ∈ s
hcc₃'' : cc₃ = t₃ • (p -ᵥ ↑((orthogonalProjection s) p)) +ᵥ cc
hcr₃' : ∀ p₁ ∈ ps, dist p₁ ↑⟨cc, hcc₃'⟩ = cr
p0 : P
hp0 : p0 ∈ ps
h' : ↑⟨cc, hcc₃'⟩ = cc
⊢ dist p0 cc * dist p0 cc +
      dist (t₃ • (p -ᵥ ↑((orthogonalProjection s) p)) +ᵥ cc) cc *
        dist (t₃ • (p -ᵥ ↑((orthogonalProjection s) p)) +ᵥ cc) cc =
    cr * cr + t₃ * y * (t₃ * y)
Module: Mathlib.Geometry.Euclidean.Incenter

Goal state before:
pp: 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
n : ℕ
inst✝¹ : NeZero n
s : Simplex ℝ P n
inst✝ : n.AtLeastTwo
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
hij : i = j
⊢ s.excenter signs ∉ line[ℝ, s.points i, s.points j]

Tactic:
simp only [hij, Set.mem_singleton_iff, Set.insert_eq_of_mem,
      AffineSubspace.mem_affineSpan_singleton]
    exact h.excenter_ne_point j

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

Goal state before:
pp: case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hpi : angle x y ≠ π
hxy : ¬x = y
hx0 : ¬x = 0
hy0 : ¬y = 0
hyx : ¬‖y‖ = ‖x‖
h : ⟪x, y⟫ / (‖x‖ * ‖y‖) = -1
⊢ False

Tactic:
real_inner_div_norm_mul_norm_eq_neg_one_iff,

Goal state after:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hpi : angle x y ≠ π
hxy : ¬x = y
hx0 : ¬x = 0
hy0 : ¬y = 0
hyx : ¬‖y‖ = ‖x‖
h : x ≠ 0 ∧ ∃ r < 0, y = r • x
⊢ False
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 p₃ : P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
hp₃ : p₃ ∈ s
hp₁p₂ : p₁ ≠ p₂
hp₁p₃ : p₁ ≠ p₃
hp₂p₃ : p₂ ≠ p₃
⊢ |(∡ p₃ p₁ s.center).toReal| ≤ π / 2

Tactic:
exact (abs_oangle_center_right_toReal_lt_pi_div_two hp₁ hp₃).le

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
⊢ dist p₁ p₃ * dist p₁ p₃ =
    dist p₁ p₂ * dist p₁ p₂ + dist p₃ p₂ * dist p₃ p₂ - 2 * dist p₁ p₂ * dist p₃ p₂ * Real.cos (∠ p₁ p₂ p₃)

Tactic:
rw [dist_eq_norm_vsub V p₁ p₃, dist_eq_norm_vsub V p₁ p₂, dist_eq_norm_vsub V p₃ p₂]
  unfold angle
  convert!
    norm_sub_sq_eq_norm_sq_add_norm_sq_sub_two_mul_norm_mul_norm_mul_cos_angle (p₁ -ᵥ p₂ : V)
      (p₃ -ᵥ p₂ : V)
  · exact (vsub_sub_vsub_cancel_right p₁ p₃ p₂).symm
  · exact (vsub_sub_vsub_cancel_right p₁ p₃ p₂).symm

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))
⊢ Finset.centroid ℝ fs s.points =
    (affineCombination ℝ univ s.pointsWithCircumcenter) (centroidWeightsWithCircumcenter fs)

Tactic:
simp_rw [centroid_def, affineCombination_apply, weightedVSubOfPoint_apply,
    sum_pointsWithCircumcenter, centroidWeightsWithCircumcenter,
    pointsWithCircumcenter_point, zero_smul, add_zero, centroidWeights,
    ← sum_indicator_subset_of_eq_zero (Function.const (Fin (n + 1)) (#fs : ℝ)⁻¹)
      (fun i wi => wi • (s.points i -ᵥ Classical.choice AddTorsor.nonempty)) fs.subset_univ fun _ =>
      zero_smul ℝ _,
    Set.indicator_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
fs : Finset (Fin (n + 1))
⊢ ∑ x, (if x ∈ ↑fs then Function.const (Fin (n + 1)) (↑(#fs))⁻¹ x else 0) • (s.points x -ᵥ Classical.choice ⋯) +ᵥ
      Classical.choice ⋯ =
    ∑ i, (if i ∈ fs then (↑(#fs))⁻¹ else 0) • (s.points i -ᵥ Classical.choice ⋯) +ᵥ Classical.choice ⋯
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))
r : ℝ
w : Fin (n + 1) → ℝ
h1 : ∑ i, w i = 1
hp : (Finset.affineCombination ℝ Finset.univ s.points) w ∈ affineSpan ℝ (Set.range s.points)
h :
  ∀ (i : Fin (n + 1)),
    (s.signedInfDist i) ((Finset.affineCombination ℝ Finset.univ s.points) w) = (if i ∈ signs then -1 else 1) * r
i : Fin (n + 1)
h' : w i = (if i ∈ signs then -1 else 1) * r / s.height i
⊢ w i = r * ((if i ∈ signs then -1 else 1) * (s.height i)⁻¹)

Tactic:
h',

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
signs : Finset (Fin (n + 1))
r : ℝ
w : Fin (n + 1) → ℝ
h1 : ∑ i, w i = 1
hp : (Finset.affineCombination ℝ Finset.univ s.points) w ∈ affineSpan ℝ (Set.range s.points)
h :
  ∀ (i : Fin (n + 1)),
    (s.signedInfDist i) ((Finset.affineCombination ℝ Finset.univ s.points) w) = (if i ∈ signs then -1 else 1) * r
i : Fin (n + 1)
h' : w i = (if i ∈ signs then -1 else 1) * r / s.height i
⊢ (if i ∈ signs then -1 else 1) * r / s.height i = r * ((if i ∈ signs then -1 else 1) * (s.height i)⁻¹)