-- Mathlib/Geometry/Euclidean/Triangle.lean
theorem sin_angle_sub_add_angle_sub_rev_eq_sin_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.sin (angle x (x - y) + angle y (y - x)) = Real.sin (angle x y)

Proof state:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
hxyn : ‖x - y‖ ≠ 0
⊢ (Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) +
              inner x (x - y) / (‖x‖ * ‖x - y‖) * Real.sin (angle y (y - x))) *
            ‖x‖ *
          ‖y‖ *
        ‖x - y‖ *
      ‖x - y‖ =
    Real.sin (angle x y) * ‖x‖ * ‖y‖ * ‖x - y‖ * ‖x - y‖

Tactic:
have H1 :
  Real.sin (angle x (x - y)) * (⟪y, y - x⟫ / (‖y‖ * ‖y - x‖)) * ‖x‖ * ‖y‖ * ‖x - y‖ =
    Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) * (⟪y, y - x⟫ / (‖y‖ * ‖y - x‖)) * ‖y‖ := by
  ring

Result:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
hxyn : ‖x - y‖ ≠ 0
H1 :
  Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) * ‖x‖ * ‖y‖ * ‖x - y‖ =
    Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) * ‖y‖
⊢ (Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) +
              inner x (x - y) / (‖x‖ * ‖x - y‖) * Real.sin (angle y (y - x))) *
            ‖x‖ *
          ‖y‖ *
        ‖x - y‖ *
      ‖x - y‖ =
    Real.sin (angle x y) * ‖x‖ * ‖y‖ * ‖x - y‖ * ‖x - y‖

Proof state:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
hxyn : ‖x - y‖ ≠ 0
⊢ Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) * ‖x‖ * ‖y‖ * ‖x - y‖ =
    Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) * ‖y‖

Tactic:
ring

Result:
no goals

Proof state:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
⊢ Real.sin (angle x (x - y) + angle y (y - x)) = Real.sin (angle x y)

Tactic:
rw [Real.sin_add, cos_angle, cos_angle]

Result:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
⊢ Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) +
      inner x (x - y) / (‖x‖ * ‖x - y‖) * Real.sin (angle y (y - x)) =
    Real.sin (angle x y)

Proof state:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
hxyn : ‖x - y‖ ≠ 0
H1 :
  Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) * ‖x‖ * ‖y‖ * ‖x - y‖ =
    Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) * ‖y‖
H2 :
  inner x (x - y) / (‖x‖ * ‖y - x‖) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖y - x‖ =
    inner x (x - y) / (‖x‖ * ‖y - x‖) * (Real.sin (angle y (y - x)) * (‖y‖ * ‖y - x‖)) * ‖x‖
H3 :
  inner x x * (inner x x - inner x y - (inner x y - inner y y)) - (inner x x - inner x y) * (inner x x - inner x y) =
    inner x x * inner y y - inner x y * inner x y
⊢ inner y y * (inner y y - inner x y - (inner x y - inner x x)) - (inner y y - inner x y) * (inner y y - inner x y) =
    inner x x * inner y y - inner x y * inner x y

Tactic:
ring

Result:
no goals

Proof state:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
hxyn : ‖x - y‖ ≠ 0
⊢ (Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) +
            inner x (x - y) / (‖x‖ * ‖x - y‖) * Real.sin (angle y (y - x))) *
          ‖x‖ *
        ‖y‖ *
      ‖x - y‖ =
    Real.sin (angle x y) * ‖x‖ * ‖y‖ * ‖x - y‖

Tactic:
apply mul_right_cancel₀ hxyn

Result:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
hxyn : ‖x - y‖ ≠ 0
⊢ (Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) +
              inner x (x - y) / (‖x‖ * ‖x - y‖) * Real.sin (angle y (y - x))) *
            ‖x‖ *
          ‖y‖ *
        ‖x - y‖ *
      ‖x - y‖ =
    Real.sin (angle x y) * ‖x‖ * ‖y‖ * ‖x - y‖ * ‖x - y‖

Proof state:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
hxyn : ‖x - y‖ ≠ 0
⊢ (Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) +
          inner x (x - y) / (‖x‖ * ‖x - y‖) * Real.sin (angle y (y - x))) *
        ‖x‖ *
      ‖y‖ =
    Real.sin (angle x y) * ‖x‖ * ‖y‖

Tactic:
apply mul_right_cancel₀ hxyn

Result:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
hxyn : ‖x - y‖ ≠ 0
⊢ (Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) +
            inner x (x - y) / (‖x‖ * ‖x - y‖) * Real.sin (angle y (y - x))) *
          ‖x‖ *
        ‖y‖ *
      ‖x - y‖ =
    Real.sin (angle x y) * ‖x‖ * ‖y‖ * ‖x - y‖

Proof state:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
hxyn : ‖x - y‖ ≠ 0
H1 :
  Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) * ‖x‖ * ‖y‖ * ‖x - y‖ =
    Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) * ‖y‖
⊢ (Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) +
              inner x (x - y) / (‖x‖ * ‖x - y‖) * Real.sin (angle y (y - x))) *
            ‖x‖ *
          ‖y‖ *
        ‖x - y‖ *
      ‖x - y‖ =
    Real.sin (angle x y) * ‖x‖ * ‖y‖ * ‖x - y‖ * ‖x - y‖

Tactic:
have H2 :
  ⟪x, x - y⟫ / (‖x‖ * ‖y - x‖) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖y - x‖ =
    ⟪x, x - y⟫ / (‖x‖ * ‖y - x‖) * (Real.sin (angle y (y - x)) * (‖y‖ * ‖y - x‖)) * ‖x‖ := by
  ring

Result:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
hxyn : ‖x - y‖ ≠ 0
H1 :
  Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) * ‖x‖ * ‖y‖ * ‖x - y‖ =
    Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) * ‖y‖
H2 :
  inner x (x - y) / (‖x‖ * ‖y - x‖) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖y - x‖ =
    inner x (x - y) / (‖x‖ * ‖y - x‖) * (Real.sin (angle y (y - x)) * (‖y‖ * ‖y - x‖)) * ‖x‖
⊢ (Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) +
              inner x (x - y) / (‖x‖ * ‖x - y‖) * Real.sin (angle y (y - x))) *
            ‖x‖ *
          ‖y‖ *
        ‖x - y‖ *
      ‖x - y‖ =
    Real.sin (angle x y) * ‖x‖ * ‖y‖ * ‖x - y‖ * ‖x - y‖

Proof state:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
⊢ Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) +
      inner x (x - y) / (‖x‖ * ‖x - y‖) * Real.sin (angle y (y - x)) =
    Real.sin (angle x y)

Tactic:
have hxyn : ‖x - y‖ ≠ 0 := fun h => hxy (eq_of_sub_eq_zero (norm_eq_zero.1 h))

Result:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
hxyn : ‖x - y‖ ≠ 0
⊢ Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) +
      inner x (x - y) / (‖x‖ * ‖x - y‖) * Real.sin (angle y (y - x)) =
    Real.sin (angle x y)