Theorem: EuclideanGeometry.cospherical_of_two_zsmul_oangle_eq_of_not_collinear
File: Mathlib/Geometry/Euclidean/Angle/Sphere.lean

Proof steps:
state_before: 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✝ : Module.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}
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✝ : Module.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}
⊢ Cospherical {p₁, p₂, p₃, p₄}
tactic: have hn' : ¬Collinear ℝ ({p₁, p₃, p₄} : Set P) := by
    rwa [← collinear_iff_of_two_zsmul_oangle_eq h]
state_before: 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✝ : Module.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}
⊢ Cospherical {p₁, p₂, p₃, p₄}
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✝ : Module.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 := ⋯ }
⊢ Cospherical {p₁, p₂, p₃, p₄}
tactic: let t₁ : Affine.Triangle ℝ P := ⟨![p₁, p₂, p₄], affineIndependent_iff_not_collinear_set.2 hn⟩
state_before: 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✝ : Module.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 := ⋯ }
⊢ Cospherical {p₁, p₂, p₃, p₄}
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✝ : Module.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 := ⋯ }
⊢ Cospherical {p₁, p₂, p₃, p₄}
tactic: let t₂ : Affine.Triangle ℝ P := ⟨![p₁, p₃, p₄], affineIndependent_iff_not_collinear_set.2 hn'⟩
state_before: 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✝ : Module.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 := ⋯ }
⊢ Cospherical {p₁, p₂, p₃, p₄}
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✝ : Module.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 := ⋯ }
⊢ ∃ s, {p₁, p₂, p₃, p₄} ⊆ Metric.sphere s.center s.radius
tactic: rw [cospherical_iff_exists_sphere]
state_before: 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✝ : Module.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 := ⋯ }
⊢ ∃ s, {p₁, p₂, p₃, p₄} ⊆ Metric.sphere s.center s.radius
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✝ : Module.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 := ⋯ }
⊢ {p₁, p₂, p₃, p₄} ⊆ Metric.sphere (Affine.Simplex.circumsphere t₂).center (Affine.Simplex.circumsphere t₂).radius
tactic: refine ⟨t₂.circumsphere, ?_⟩
state_before: 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✝ : Module.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 := ⋯ }
⊢ {p₁, p₂, p₃, p₄} ⊆ Metric.sphere (Affine.Simplex.circumsphere t₂).center (Affine.Simplex.circumsphere t₂).radius
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✝ : Module.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 := ⋯ }
⊢ p₁ ∈ Metric.sphere (Affine.Simplex.circumsphere t₂).center (Affine.Simplex.circumsphere t₂).radius ∧
    p₂ ∈ Metric.sphere (Affine.Simplex.circumsphere t₂).center (Affine.Simplex.circumsphere t₂).radius ∧
      p₃ ∈ Metric.sphere (Affine.Simplex.circumsphere t₂).center (Affine.Simplex.circumsphere t₂).radius ∧
        p₄ ∈ Metric.sphere (Affine.Simplex.circumsphere t₂).center (Affine.Simplex.circumsphere t₂).radius
tactic: simp_rw [Set.insert_subset_iff, Set.singleton_subset_iff]
state_before: 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✝ : Module.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 := ⋯ }
⊢ p₁ ∈ Metric.sphere (Affine.Simplex.circumsphere t₂).center (Affine.Simplex.circumsphere t₂).radius ∧
    p₂ ∈ Metric.sphere (Affine.Simplex.circumsphere t₂).center (Affine.Simplex.circumsphere t₂).radius ∧
      p₃ ∈ Metric.sphere (Affine.Simplex.circumsphere t₂).center (Affine.Simplex.circumsphere t₂).radius ∧
        p₄ ∈ Metric.sphere (Affine.Simplex.circumsphere t₂).center (Affine.Simplex.circumsphere t₂).radius
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✝ : Module.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 := ⋯ }
⊢ p₂ ∈ Metric.sphere (Affine.Simplex.circumsphere t₂).center (Affine.Simplex.circumsphere t₂).radius
tactic: refine ⟨t₂.mem_circumsphere 0, ?_, t₂.mem_circumsphere 1, t₂.mem_circumsphere 2⟩
state_before: 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✝ : Module.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 := ⋯ }
⊢ p₂ ∈ Metric.sphere (Affine.Simplex.circumsphere t₂).center (Affine.Simplex.circumsphere t₂).radius
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✝ : Module.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 := ⋯ }
⊢ p₂ ∈ Metric.sphere (Affine.Simplex.circumsphere t₁).center (Affine.Simplex.circumsphere t₁).radius
tactic: rw [Affine.Triangle.circumsphere_eq_circumsphere_of_eq_of_eq_of_two_zsmul_oangle_eq
    (by decide : (0 : Fin 3) ≠ 1) (by decide : (0 : Fin 3) ≠ 2) (by decide)
    (show t₂.points 0 = t₁.points 0 from rfl) rfl h.symm]
state_before: 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✝ : Module.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 := ⋯ }
⊢ p₂ ∈ Metric.sphere (Affine.Simplex.circumsphere t₁).center (Affine.Simplex.circumsphere t₁).radius
state_after: no goals
tactic: exact t₁.mem_circumsphere 1