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) = ↑(Real.arcsin (‖y‖ / ‖x + y‖))

Tactic:
by

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

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫ = 0
⊢ Real.cos (angle x (x - y)) * ‖x - y‖ = ‖x‖

Tactic:
by

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

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫ = 0
⊢ Real.sin (angle x (x - y)) * ‖x - y‖ = ‖y‖

Tactic:
by

Goal state after:
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 + y) y = ↑(Real.arcsin (‖x‖ / ‖x + y‖))

Tactic:
by

Goal state after:
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 p₅ p₆ : P
h₂ : p₂ ∉ line[ℝ, p₁, p₃]
h₄ : p₄ ∈ line[ℝ, p₁, p₃]
h₆ : p₆ ∈ line[ℝ, p₁, p₃]
h₁₂₄₅ : line[ℝ, p₁, p₂] ∥ line[ℝ, p₄, p₅]
h₃₂₆₅ : line[ℝ, p₃, p₂] ∥ line[ℝ, p₆, p₅]
⊢ ∡ p₁ p₂ p₃ = ∡ p₄ p₅ p₆

Tactic:
by

Goal state after:
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)
p p₁ p₂ p₃ : P
hp₁ : p₁ ≠ p
hp₂ : p₂ ≠ p
hp₃ : p₃ ≠ p
⊢ ∡ p₁ p p₂ + ∡ p₂ p p₃ + ∡ p₃ p p₁ = 0

Tactic:
by

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

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫ = 0
h0 : x ≠ 0 ∨ y = 0
⊢ Real.tan (angle x (x - y)) * ‖x‖ = ‖y‖

Tactic:
by

Goal state after:
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) = ↑(Real.arctan (‖y‖ / ‖x‖))

Tactic:
by

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

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫ = 0
h0 : x ≠ 0 ∨ y = 0
⊢ ‖x‖ / Real.cos (angle x (x - y)) = ‖x - y‖

Tactic:
by

Goal state after:
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 + y) y = ↑(Real.arctan (‖x‖ / ‖y‖))

Tactic:
by

Goal state after:
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 : P
h : dist p₁ p₂ = dist p₁ p₃
⊢ ∡ p₁ p₂ p₃ = ∡ p₂ p₃ p₁

Tactic:
by

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

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫ = 0
h0 : x = 0 ∨ y ≠ 0
⊢ ‖y‖ / Real.sin (angle x (x - y)) = ‖x - y‖

Tactic:
by

Goal state after:
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‖ / ‖x + y‖

Tactic:
by

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

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫ = 0
h0 : x = 0 ∨ y ≠ 0
⊢ ‖y‖ / Real.tan (angle x (x - y)) = ‖x‖

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Basic

Goal state before:
pp: V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
w x y z : V
h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z
hs : (o.oangle w x).sign = (o.oangle y z).sign
⊢ o.oangle w x = o.oangle y z

Tactic:
by

Goal state after:
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
⊢ dist p₁ p₃ * dist p₁ p₃ = dist p₁ p₂ * dist p₁ p₂ + dist p₃ p₂ * dist p₃ p₂ ↔ ∠ p₁ p₂ p₃ = π / 2

Tactic:
by

Goal state after:
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 + y) y).cos = ‖y‖ / ‖x + y‖

Tactic:
by

Goal state after:
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 : P
hn : p₂ ≠ p₃
h : dist p₁ p₂ = dist p₁ p₃
⊢ ∡ p₃ p₁ p₂ = ↑π - 2 • ∡ p₁ p₂ p₃

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Basic

Goal state before:
pp: V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
w x y z : V
hw : w ≠ 0
hx : x ≠ 0
hy : y ≠ 0
hz : z ≠ 0
hs : (o.oangle w x).sign = (o.oangle y z).sign
⊢ InnerProductGeometry.angle w x = InnerProductGeometry.angle y z ↔ o.oangle w x = o.oangle y z

Tactic:
by

Goal state after:
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
⊢ ∠ p₂ p₃ p₁ = Real.arccos (dist p₃ p₂ / dist p₁ p₃)

Tactic:
by

Goal state after:
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 : P
h : dist p₁ p₂ = dist p₁ p₃
⊢ |(∡ p₁ p₂ p₃).toReal| < π / 2

Tactic:
by

Goal state after:
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)
p p₁ p₂ : P
h : (∡ p₁ p p₂).sign = 0
⊢ p₁ = p ∨ p₂ = p ∨ ∠ p₁ p p₂ = 0 ∨ ∠ p₁ p p₂ = π

Tactic:
by

Goal state after:
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 = ‖y‖ / ‖x + y‖

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Basic

Goal state before:
pp: V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
w x y z : V
h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z
hs : (o.oangle w x).sign = -(o.oangle y z).sign
⊢ o.oangle w x = -o.oangle y z

Tactic:
by

Goal state after:
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
⊢ ∠ p₂ p₃ p₁ = Real.arcsin (dist p₁ p₂ / dist p₁ p₃)

Tactic:
by

Goal state after:
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 + y) y).sin = ‖x‖ / ‖x + y‖

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Basic

Goal state before:
pp: V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
w x y z : V
hw : w ≠ 0
hx : x ≠ 0
hy : y ≠ 0
hz : z ≠ 0
hs : (o.oangle w x).sign = -(o.oangle y z).sign
⊢ InnerProductGeometry.angle w x = InnerProductGeometry.angle y z ↔ o.oangle w x = -o.oangle y z

Tactic:
by

Goal state after:
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:
by

Goal state after:
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 : p₃ ≠ p₂
⊢ ∠ p₂ p₃ p₁ = Real.arctan (dist p₁ p₂ / dist p₃ p₂)

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Basic

Goal state before:
pp: V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (o.oangle x y).sign = 1
⊢ o.oangle x y = ↑(InnerProductGeometry.angle x y)

Tactic:
by

Goal state after:
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 + y) y).tan = ‖x‖ / ‖y‖

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Basic

Goal state before:
pp: V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (o.oangle x y).sign = -1
⊢ o.oangle x y = -↑(InnerProductGeometry.angle x y)

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Basic

Goal state before:
pp: case refine_2
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
hx : x ≠ 0
hy : y ≠ 0
h : InnerProductGeometry.angle x y = 0
⊢ o.oangle x y = 0

Tactic:
have ha := o.oangle_eq_angle_or_eq_neg_angle hx hy

Goal state after:
case refine_2
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
hx : x ≠ 0
hy : y ≠ 0
h : InnerProductGeometry.angle x y = 0
ha : o.oangle x y = ↑(InnerProductGeometry.angle x y) ∨ o.oangle x y = -↑(InnerProductGeometry.angle x y)
⊢ o.oangle x y = 0
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
⊢ 0 < ∠ p₂ p₃ p₁

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Basic

Goal state before:
pp: V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
hx : x ≠ 0
hy : y ≠ 0
⊢ o.oangle x y = 0 ↔ InnerProductGeometry.angle x y = 0

Tactic:
by

Goal state after:
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:
by

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Basic

Goal state before:
pp: case neg.refine_2
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
hx : ¬x = 0
hy : ¬y = 0
h : InnerProductGeometry.angle x y = π
⊢ o.oangle x y = ↑π

Tactic:
have ha := o.oangle_eq_angle_or_eq_neg_angle hx hy

Goal state after:
case neg.refine_2
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
hx : ¬x = 0
hy : ¬y = 0
h : InnerProductGeometry.angle x y = π
ha : o.oangle x y = ↑(InnerProductGeometry.angle x y) ∨ o.oangle x y = -↑(InnerProductGeometry.angle x y)
⊢ o.oangle x y = ↑π
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
⊢ ∠ p₂ p₃ p₁ ≤ π / 2

Tactic:
by

Goal state after:
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 + y) y).cos * ‖x + y‖ = ‖y‖

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Basic

Goal state before:
pp: V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
⊢ o.oangle x y = ↑π ↔ InnerProductGeometry.angle x y = π

Tactic:
by

Goal state after:
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 p₅ p₆ : P
h : ∠ p₁ p₂ p₃ = ∠ p₄ p₅ p₆
h1 : p₂ ≠ p₁
h2 : p₂ ≠ p₃
h3 : p₅ ≠ p₄
h4 : p₅ ≠ p₆
⊢ ∡ p₁ p₂ p₃ = ∡ p₄ p₅ p₆ ∨ ∡ p₁ p₂ p₃ = -∡ p₄ p₅ p₆

Tactic:
by

Goal state after:
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 : P
h : ∡ p₁ p₂ p₃ = ↑(π / 2)
⊢ ∠ p₁ p₂ p₃ = π / 2

Tactic:
by

Goal state after:
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:
by

Goal state after:
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 : p₃ ≠ p₂
⊢ ∠ p₂ p₃ p₁ < π / 2

Tactic:
by

Goal state after:
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 : P
h : ∡ p₁ p₂ p₃ = ↑(π / 2)
⊢ ∠ p₃ p₂ p₁ = π / 2

Tactic:
by

Goal state after:
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 : P
h : ∡ p₁ p₂ p₃ = ↑(-π / 2)
⊢ ∠ p₁ p₂ p₃ = π / 2

Tactic:
by

Goal state after:
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 + y) y).sin * ‖x + y‖ = ‖x‖

Tactic:
by

Goal state after:
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 : P
h : ∡ p₁ p₂ p₃ = ↑(-π / 2)
⊢ ∠ p₃ p₂ p₁ = π / 2

Tactic:
by

Goal state after:
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:
by

Goal state after:
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 + y) y).tan * ‖y‖ = ‖x‖

Tactic:
by

Goal state after:
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
⊢ Real.cos (∠ p₂ p₃ p₁) = dist p₃ p₂ / dist p₁ p₃

Tactic:
by

Goal state after:
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 : P
⊢ -(∡ p₁ p₂ p₃).sign = (∡ p₂ p₁ p₃).sign

Tactic:
by

Goal state after:
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:
by

Goal state after:
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 : P
⊢ -(∡ p₁ p₂ p₃).sign = (∡ p₃ p₂ p₁).sign

Tactic:
by

Goal state after:
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 : P
⊢ -(∡ p₁ p₂ p₃).sign = (∡ p₁ p₃ p₂).sign

Tactic:
by

Goal state after:
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 : P
⊢ (∡ p₂ p₃ p₁).sign = (∡ p₁ p₂ p₃).sign

Tactic:
by

Goal state after:
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 + y) y).cos = ‖x + y‖

Tactic:
by

Goal state after:
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 : P
⊢ ∡ p₁ p₂ p₃ = ↑π ↔ Sbtw ℝ p₁ p₂ p₃

Tactic:
by

Goal state after:
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 : P
h : Sbtw ℝ p₁ p₂ p₃
⊢ ∡ p₃ p₂ p₁ = ↑π

Tactic:
by

Goal state after:
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:
by

Goal state after:
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 + y) y).sin = ‖x + y‖

Tactic:
by

Goal state after:
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 : P
h : Wbtw ℝ p₁ p₂ p₃
⊢ ∡ p₂ p₁ p₃ = 0

Tactic:
by

Goal state after:
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 : P
h : Wbtw ℝ p₁ p₂ p₃
⊢ ∡ p₃ p₁ p₂ = 0

Tactic:
by

Goal state after:
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:
by

Goal state after:
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 + y) y).tan = ‖y‖

Tactic:
by

Goal state after:
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 : P
⊢ ∡ p₁ p₂ p₃ = 0 ↔ Wbtw ℝ p₂ p₁ p₃ ∨ Wbtw ℝ p₂ p₃ p₁

Tactic:
by

Goal state after:
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 y (y - x)).sign = 1

Tactic:
by

Goal state after:
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
⊢ Real.sin (∠ p₂ p₃ p₁) = dist p₁ p₂ / dist p₁ p₃

Tactic:
by

Goal state after: