File: Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean

full_name: Orientation.oangle_add_right_eq_arccos_of_oangle_eq_pi_div_two
code: /-- An angle in a right-angled triangle expressed using `arccos`. -/
theorem oangle_add_right_eq_arccos_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    o.oangle x (x + y) = Real.arccos (‖x‖ / ‖x + y‖) 
goal: o.oangle x (x + y) = ↑(Real.arccos (‖x‖ / ‖x + y‖))

full_name: Orientation.oangle_add_left_eq_arccos_of_oangle_eq_pi_div_two
code: /-- An angle in a right-angled triangle expressed using `arccos`. -/
theorem oangle_add_left_eq_arccos_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    o.oangle (x + y) y = Real.arccos (‖y‖ / ‖x + y‖) 
goal: o.oangle (x + y) y = ↑(Real.arccos (‖y‖ / ‖x + y‖))

full_name: Orientation.oangle_add_right_eq_arcsin_of_oangle_eq_pi_div_two
code: /-- An angle in a right-angled triangle expressed using `arcsin`. -/
theorem oangle_add_right_eq_arcsin_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    o.oangle x (x + y) = Real.arcsin (‖y‖ / ‖x + y‖) 
goal: o.oangle x (x + y) = ↑(Real.arcsin (‖y‖ / ‖x + y‖))

full_name: Orientation.oangle_add_left_eq_arcsin_of_oangle_eq_pi_div_two
code: /-- An angle in a right-angled triangle expressed using `arcsin`. -/
theorem oangle_add_left_eq_arcsin_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    o.oangle (x + y) y = Real.arcsin (‖x‖ / ‖x + y‖) 
goal: o.oangle (x + y) y = ↑(Real.arcsin (‖x‖ / ‖x + y‖))

full_name: Orientation.oangle_add_right_eq_arctan_of_oangle_eq_pi_div_two
code: /-- An angle in a right-angled triangle expressed using `arctan`. -/
theorem oangle_add_right_eq_arctan_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    o.oangle x (x + y) = Real.arctan (‖y‖ / ‖x‖) 
goal: o.oangle x (x + y) = ↑(Real.arctan (‖y‖ / ‖x‖))

full_name: Orientation.oangle_add_left_eq_arctan_of_oangle_eq_pi_div_two
code: /-- An angle in a right-angled triangle expressed using `arctan`. -/
theorem oangle_add_left_eq_arctan_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    o.oangle (x + y) y = Real.arctan (‖x‖ / ‖y‖) 
goal: o.oangle (x + y) y = ↑(Real.arctan (‖x‖ / ‖y‖))

full_name: Orientation.cos_oangle_add_right_of_oangle_eq_pi_div_two
code: /-- The cosine of an angle in a right-angled triangle as a ratio of sides. -/
theorem cos_oangle_add_right_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    Real.Angle.cos (o.oangle x (x + y)) = ‖x‖ / ‖x + y‖ 
goal: (o.oangle x (x + y)).cos = ‖x‖ / ‖x + y‖

full_name: Orientation.cos_oangle_add_left_of_oangle_eq_pi_div_two
code: /-- The cosine of an angle in a right-angled triangle as a ratio of sides. -/
theorem cos_oangle_add_left_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    Real.Angle.cos (o.oangle (x + y) y) = ‖y‖ / ‖x + y‖ 
goal: (o.oangle (x + y) y).cos = ‖y‖ / ‖x + y‖

full_name: Orientation.sin_oangle_add_right_of_oangle_eq_pi_div_two
code: /-- The sine of an angle in a right-angled triangle as a ratio of sides. -/
theorem sin_oangle_add_right_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    Real.Angle.sin (o.oangle x (x + y)) = ‖y‖ / ‖x + y‖ 
goal: (o.oangle x (x + y)).sin = ‖y‖ / ‖x + y‖

full_name: Orientation.sin_oangle_add_left_of_oangle_eq_pi_div_two
code: /-- The sine of an angle in a right-angled triangle as a ratio of sides. -/
theorem sin_oangle_add_left_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    Real.Angle.sin (o.oangle (x + y) y) = ‖x‖ / ‖x + y‖ 
goal: (o.oangle (x + y) y).sin = ‖x‖ / ‖x + y‖

full_name: Orientation.tan_oangle_add_right_of_oangle_eq_pi_div_two
code: /-- The tangent of an angle in a right-angled triangle as a ratio of sides. -/
theorem tan_oangle_add_right_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    Real.Angle.tan (o.oangle x (x + y)) = ‖y‖ / ‖x‖ 
goal: (o.oangle x (x + y)).tan = ‖y‖ / ‖x‖

full_name: Orientation.tan_oangle_add_left_of_oangle_eq_pi_div_two
code: /-- The tangent of an angle in a right-angled triangle as a ratio of sides. -/
theorem tan_oangle_add_left_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    Real.Angle.tan (o.oangle (x + y) y) = ‖x‖ / ‖y‖ 
goal: (o.oangle (x + y) y).tan = ‖x‖ / ‖y‖

full_name: Orientation.cos_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two
code: /-- The cosine of an angle in a right-angled triangle multiplied by the hypotenuse equals the
adjacent side. -/
theorem cos_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : Real.Angle.cos (o.oangle x (x + y)) * ‖x + y‖ = ‖x‖ 
goal: (o.oangle x (x + y)).cos * ‖x + y‖ = ‖x‖

full_name: Orientation.cos_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two
code: /-- The cosine of an angle in a right-angled triangle multiplied by the hypotenuse equals the
adjacent side. -/
theorem cos_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : Real.Angle.cos (o.oangle (x + y) y) * ‖x + y‖ = ‖y‖ 
goal: (o.oangle (x + y) y).cos * ‖x + y‖ = ‖y‖

full_name: Orientation.sin_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two
code: /-- The sine of an angle in a right-angled triangle multiplied by the hypotenuse equals the
opposite side. -/
theorem sin_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : Real.Angle.sin (o.oangle x (x + y)) * ‖x + y‖ = ‖y‖ 
goal: (o.oangle x (x + y)).sin * ‖x + y‖ = ‖y‖

full_name: Orientation.sin_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two
code: /-- The sine of an angle in a right-angled triangle multiplied by the hypotenuse equals the
opposite side. -/
theorem sin_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : Real.Angle.sin (o.oangle (x + y) y) * ‖x + y‖ = ‖x‖ 
goal: (o.oangle (x + y) y).sin * ‖x + y‖ = ‖x‖

full_name: Orientation.tan_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two
code: /-- The tangent of an angle in a right-angled triangle multiplied by the adjacent side equals
the opposite side. -/
theorem tan_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : Real.Angle.tan (o.oangle x (x + y)) * ‖x‖ = ‖y‖ 
goal: (o.oangle x (x + y)).tan * ‖x‖ = ‖y‖

full_name: Orientation.tan_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two
code: /-- The tangent of an angle in a right-angled triangle multiplied by the adjacent side equals
the opposite side. -/
theorem tan_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : Real.Angle.tan (o.oangle (x + y) y) * ‖y‖ = ‖x‖ 
goal: (o.oangle (x + y) y).tan * ‖y‖ = ‖x‖

full_name: Orientation.norm_div_cos_oangle_add_right_of_oangle_eq_pi_div_two
code: /-- A side of a right-angled triangle divided by the cosine of the adjacent angle equals the
hypotenuse. -/
theorem norm_div_cos_oangle_add_right_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : ‖x‖ / Real.Angle.cos (o.oangle x (x + y)) = ‖x + y‖ 
goal: ‖x‖ / (o.oangle x (x + y)).cos = ‖x + y‖

full_name: Orientation.norm_div_cos_oangle_add_left_of_oangle_eq_pi_div_two
code: /-- A side of a right-angled triangle divided by the cosine of the adjacent angle equals the
hypotenuse. -/
theorem norm_div_cos_oangle_add_left_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : ‖y‖ / Real.Angle.cos (o.oangle (x + y) y) = ‖x + y‖ 
goal: ‖y‖ / (o.oangle (x + y) y).cos = ‖x + y‖

full_name: Orientation.norm_div_sin_oangle_add_right_of_oangle_eq_pi_div_two
code: /-- A side of a right-angled triangle divided by the sine of the opposite angle equals the
hypotenuse. -/
theorem norm_div_sin_oangle_add_right_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : ‖y‖ / Real.Angle.sin (o.oangle x (x + y)) = ‖x + y‖ 
goal: ‖y‖ / (o.oangle x (x + y)).sin = ‖x + y‖

full_name: Orientation.norm_div_sin_oangle_add_left_of_oangle_eq_pi_div_two
code: /-- A side of a right-angled triangle divided by the sine of the opposite angle equals the
hypotenuse. -/
theorem norm_div_sin_oangle_add_left_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : ‖x‖ / Real.Angle.sin (o.oangle (x + y) y) = ‖x + y‖ 
goal: ‖x‖ / (o.oangle (x + y) y).sin = ‖x + y‖

full_name: Orientation.norm_div_tan_oangle_add_right_of_oangle_eq_pi_div_two
code: /-- A side of a right-angled triangle divided by the tangent of the opposite angle equals the
adjacent side. -/
theorem norm_div_tan_oangle_add_right_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : ‖y‖ / Real.Angle.tan (o.oangle x (x + y)) = ‖x‖ 
goal: ‖y‖ / (o.oangle x (x + y)).tan = ‖x‖

full_name: Orientation.norm_div_tan_oangle_add_left_of_oangle_eq_pi_div_two
code: /-- A side of a right-angled triangle divided by the tangent of the opposite angle equals the
adjacent side. -/
theorem norm_div_tan_oangle_add_left_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : ‖x‖ / Real.Angle.tan (o.oangle (x + y) y) = ‖y‖ 
goal: ‖x‖ / (o.oangle (x + y) y).tan = ‖y‖

full_name: Orientation.oangle_sub_right_eq_arccos_of_oangle_eq_pi_div_two
code: /-- An angle in a right-angled triangle expressed using `arccos`, version subtracting vectors. -/
theorem oangle_sub_right_eq_arccos_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    o.oangle y (y - x) = Real.arccos (‖y‖ / ‖y - x‖) 
goal: o.oangle y (y - x) = ↑(Real.arccos (‖y‖ / ‖y - x‖))

full_name: Orientation.oangle_sub_left_eq_arccos_of_oangle_eq_pi_div_two
code: /-- An angle in a right-angled triangle expressed using `arccos`, version subtracting vectors. -/
theorem oangle_sub_left_eq_arccos_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    o.oangle (x - y) x = Real.arccos (‖x‖ / ‖x - y‖) 
goal: o.oangle (x - y) x = ↑(Real.arccos (‖x‖ / ‖x - y‖))

full_name: Orientation.oangle_sub_right_eq_arcsin_of_oangle_eq_pi_div_two
code: /-- An angle in a right-angled triangle expressed using `arcsin`, version subtracting vectors. -/
theorem oangle_sub_right_eq_arcsin_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    o.oangle y (y - x) = Real.arcsin (‖x‖ / ‖y - x‖) 
goal: o.oangle y (y - x) = ↑(Real.arcsin (‖x‖ / ‖y - x‖))

full_name: Orientation.oangle_sub_left_eq_arcsin_of_oangle_eq_pi_div_two
code: /-- An angle in a right-angled triangle expressed using `arcsin`, version subtracting vectors. -/
theorem oangle_sub_left_eq_arcsin_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    o.oangle (x - y) x = Real.arcsin (‖y‖ / ‖x - y‖) 
goal: o.oangle (x - y) x = ↑(Real.arcsin (‖y‖ / ‖x - y‖))

full_name: Orientation.oangle_sub_right_eq_arctan_of_oangle_eq_pi_div_two
code: /-- An angle in a right-angled triangle expressed using `arctan`, version subtracting vectors. -/
theorem oangle_sub_right_eq_arctan_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    o.oangle y (y - x) = Real.arctan (‖x‖ / ‖y‖) 
goal: o.oangle y (y - x) = ↑(Real.arctan (‖x‖ / ‖y‖))

full_name: Orientation.oangle_sub_left_eq_arctan_of_oangle_eq_pi_div_two
code: /-- An angle in a right-angled triangle expressed using `arctan`, version subtracting vectors. -/
theorem oangle_sub_left_eq_arctan_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    o.oangle (x - y) x = Real.arctan (‖y‖ / ‖x‖) 
goal: o.oangle (x - y) x = ↑(Real.arctan (‖y‖ / ‖x‖))

full_name: Orientation.cos_oangle_sub_right_of_oangle_eq_pi_div_two
code: /-- The cosine of an angle in a right-angled triangle as a ratio of sides, version subtracting
vectors. -/
theorem cos_oangle_sub_right_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    Real.Angle.cos (o.oangle y (y - x)) = ‖y‖ / ‖y - x‖ 
goal: (o.oangle y (y - x)).cos = ‖y‖ / ‖y - x‖

full_name: Orientation.cos_oangle_sub_left_of_oangle_eq_pi_div_two
code: /-- The cosine of an angle in a right-angled triangle as a ratio of sides, version subtracting
vectors. -/
theorem cos_oangle_sub_left_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    Real.Angle.cos (o.oangle (x - y) x) = ‖x‖ / ‖x - y‖ 
goal: (o.oangle (x - y) x).cos = ‖x‖ / ‖x - y‖

full_name: Orientation.sin_oangle_sub_right_of_oangle_eq_pi_div_two
code: /-- The sine of an angle in a right-angled triangle as a ratio of sides, version subtracting
vectors. -/
theorem sin_oangle_sub_right_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    Real.Angle.sin (o.oangle y (y - x)) = ‖x‖ / ‖y - x‖ 
goal: (o.oangle y (y - x)).sin = ‖x‖ / ‖y - x‖

full_name: Orientation.sin_oangle_sub_left_of_oangle_eq_pi_div_two
code: /-- The sine of an angle in a right-angled triangle as a ratio of sides, version subtracting
vectors. -/
theorem sin_oangle_sub_left_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    Real.Angle.sin (o.oangle (x - y) x) = ‖y‖ / ‖x - y‖ 
goal: (o.oangle (x - y) x).sin = ‖y‖ / ‖x - y‖

full_name: Orientation.tan_oangle_sub_right_of_oangle_eq_pi_div_two
code: /-- The tangent of an angle in a right-angled triangle as a ratio of sides, version subtracting
vectors. -/
theorem tan_oangle_sub_right_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    Real.Angle.tan (o.oangle y (y - x)) = ‖x‖ / ‖y‖ 
goal: (o.oangle y (y - x)).tan = ‖x‖ / ‖y‖

full_name: Orientation.tan_oangle_sub_left_of_oangle_eq_pi_div_two
code: /-- The tangent of an angle in a right-angled triangle as a ratio of sides, version subtracting
vectors. -/
theorem tan_oangle_sub_left_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    Real.Angle.tan (o.oangle (x - y) x) = ‖y‖ / ‖x‖ 
goal: (o.oangle (x - y) x).tan = ‖y‖ / ‖x‖

full_name: Orientation.cos_oangle_sub_right_mul_norm_of_oangle_eq_pi_div_two
code: /-- The cosine of an angle in a right-angled triangle multiplied by the hypotenuse equals the
adjacent side, version subtracting vectors. -/
theorem cos_oangle_sub_right_mul_norm_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : Real.Angle.cos (o.oangle y (y - x)) * ‖y - x‖ = ‖y‖ 
goal: (o.oangle y (y - x)).cos * ‖y - x‖ = ‖y‖

full_name: Orientation.cos_oangle_sub_left_mul_norm_of_oangle_eq_pi_div_two
code: /-- The cosine of an angle in a right-angled triangle multiplied by the hypotenuse equals the
adjacent side, version subtracting vectors. -/
theorem cos_oangle_sub_left_mul_norm_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : Real.Angle.cos (o.oangle (x - y) x) * ‖x - y‖ = ‖x‖ 
goal: (o.oangle (x - y) x).cos * ‖x - y‖ = ‖x‖

full_name: Orientation.sin_oangle_sub_right_mul_norm_of_oangle_eq_pi_div_two
code: /-- The sine of an angle in a right-angled triangle multiplied by the hypotenuse equals the
opposite side, version subtracting vectors. -/
theorem sin_oangle_sub_right_mul_norm_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : Real.Angle.sin (o.oangle y (y - x)) * ‖y - x‖ = ‖x‖ 
goal: (o.oangle y (y - x)).sin * ‖y - x‖ = ‖x‖

full_name: Orientation.sin_oangle_sub_left_mul_norm_of_oangle_eq_pi_div_two
code: /-- The sine of an angle in a right-angled triangle multiplied by the hypotenuse equals the
opposite side, version subtracting vectors. -/
theorem sin_oangle_sub_left_mul_norm_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : Real.Angle.sin (o.oangle (x - y) x) * ‖x - y‖ = ‖y‖ 
goal: (o.oangle (x - y) x).sin * ‖x - y‖ = ‖y‖

full_name: Orientation.tan_oangle_sub_right_mul_norm_of_oangle_eq_pi_div_two
code: /-- The tangent of an angle in a right-angled triangle multiplied by the adjacent side equals
the opposite side, version subtracting vectors. -/
theorem tan_oangle_sub_right_mul_norm_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : Real.Angle.tan (o.oangle y (y - x)) * ‖y‖ = ‖x‖ 
goal: (o.oangle y (y - x)).tan * ‖y‖ = ‖x‖

full_name: Orientation.tan_oangle_sub_left_mul_norm_of_oangle_eq_pi_div_two
code: /-- The tangent of an angle in a right-angled triangle multiplied by the adjacent side equals
the opposite side, version subtracting vectors. -/
theorem tan_oangle_sub_left_mul_norm_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : Real.Angle.tan (o.oangle (x - y) x) * ‖x‖ = ‖y‖ 
goal: (o.oangle (x - y) x).tan * ‖x‖ = ‖y‖

full_name: Orientation.norm_div_cos_oangle_sub_right_of_oangle_eq_pi_div_two
code: /-- A side of a right-angled triangle divided by the cosine of the adjacent angle equals the
hypotenuse, version subtracting vectors. -/
theorem norm_div_cos_oangle_sub_right_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : ‖y‖ / Real.Angle.cos (o.oangle y (y - x)) = ‖y - x‖ 
goal: ‖y‖ / (o.oangle y (y - x)).cos = ‖y - x‖

full_name: Orientation.norm_div_cos_oangle_sub_left_of_oangle_eq_pi_div_two
code: /-- A side of a right-angled triangle divided by the cosine of the adjacent angle equals the
hypotenuse, version subtracting vectors. -/
theorem norm_div_cos_oangle_sub_left_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : ‖x‖ / Real.Angle.cos (o.oangle (x - y) x) = ‖x - y‖ 
goal: ‖x‖ / (o.oangle (x - y) x).cos = ‖x - y‖

full_name: Orientation.norm_div_sin_oangle_sub_right_of_oangle_eq_pi_div_two
code: /-- A side of a right-angled triangle divided by the sine of the opposite angle equals the
hypotenuse, version subtracting vectors. -/
theorem norm_div_sin_oangle_sub_right_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : ‖x‖ / Real.Angle.sin (o.oangle y (y - x)) = ‖y - x‖ 
goal: ‖x‖ / (o.oangle y (y - x)).sin = ‖y - x‖

full_name: Orientation.norm_div_sin_oangle_sub_left_of_oangle_eq_pi_div_two
code: /-- A side of a right-angled triangle divided by the sine of the opposite angle equals the
hypotenuse, version subtracting vectors. -/
theorem norm_div_sin_oangle_sub_left_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : ‖y‖ / Real.Angle.sin (o.oangle (x - y) x) = ‖x - y‖ 
goal: ‖y‖ / (o.oangle (x - y) x).sin = ‖x - y‖

full_name: Orientation.norm_div_tan_oangle_sub_right_of_oangle_eq_pi_div_two
code: /-- A side of a right-angled triangle divided by the tangent of the opposite angle equals the
adjacent side, version subtracting vectors. -/
theorem norm_div_tan_oangle_sub_right_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : ‖x‖ / Real.Angle.tan (o.oangle y (y - x)) = ‖y‖ 
goal: ‖x‖ / (o.oangle y (y - x)).tan = ‖y‖

full_name: Orientation.norm_div_tan_oangle_sub_left_of_oangle_eq_pi_div_two
code: /-- A side of a right-angled triangle divided by the tangent of the opposite angle equals the
adjacent side, version subtracting vectors. -/
theorem norm_div_tan_oangle_sub_left_of_oangle_eq_pi_div_two {x y : V}
    (h : o.oangle x y = ↑(π / 2)) : ‖y‖ / Real.Angle.tan (o.oangle (x - y) x) = ‖x‖ 
goal: ‖y‖ / (o.oangle (x - y) x).tan = ‖x‖

full_name: Orientation.oangle_add_right_smul_rotation_pi_div_two
code: /-- An angle in a right-angled triangle expressed using `arctan`, where one side is a multiple
of a rotation of another by `π / 2`. -/
theorem oangle_add_right_smul_rotation_pi_div_two {x : V} (h : x ≠ 0) (r : ℝ) :
    o.oangle x (x + r • o.rotation (π / 2 : ℝ) x) = Real.arctan r 
goal: o.oangle x (x + r • (o.rotation ↑(π / 2)) x) = ↑(Real.arctan r)

full_name: Orientation.oangle_add_left_smul_rotation_pi_div_two
code: /-- An angle in a right-angled triangle expressed using `arctan`, where one side is a multiple
of a rotation of another by `π / 2`. -/
theorem oangle_add_left_smul_rotation_pi_div_two {x : V} (h : x ≠ 0) (r : ℝ) :
    o.oangle (x + r • o.rotation (π / 2 : ℝ) x) (r • o.rotation (π / 2 : ℝ) x)
      = Real.arctan r⁻¹ 
goal: o.oangle (x + r • (o.rotation ↑(π / 2)) x) (r • (o.rotation ↑(π / 2)) x) = ↑(Real.arctan r⁻¹)

full_name: Orientation.tan_oangle_add_right_smul_rotation_pi_div_two
code: /-- The tangent of an angle in a right-angled triangle, where one side is a multiple of a
rotation of another by `π / 2`. -/
theorem tan_oangle_add_right_smul_rotation_pi_div_two {x : V} (h : x ≠ 0) (r : ℝ) :
    Real.Angle.tan (o.oangle x (x + r • o.rotation (π / 2 : ℝ) x)) = r 
goal: (o.oangle x (x + r • (o.rotation ↑(π / 2)) x)).tan = r

full_name: Orientation.tan_oangle_add_left_smul_rotation_pi_div_two
code: /-- The tangent of an angle in a right-angled triangle, where one side is a multiple of a
rotation of another by `π / 2`. -/
theorem tan_oangle_add_left_smul_rotation_pi_div_two {x : V} (h : x ≠ 0) (r : ℝ) :
    Real.Angle.tan (o.oangle (x + r • o.rotation (π / 2 : ℝ) x) (r • o.rotation (π / 2 : ℝ) x)) =
      r⁻¹ 
goal: (o.oangle (x + r • (o.rotation ↑(π / 2)) x) (r • (o.rotation ↑(π / 2)) x)).tan = r⁻¹

full_name: Orientation.oangle_sub_right_smul_rotation_pi_div_two
code: /-- An angle in a right-angled triangle expressed using `arctan`, where one side is a multiple
of a rotation of another by `π / 2`, version subtracting vectors. -/
theorem oangle_sub_right_smul_rotation_pi_div_two {x : V} (h : x ≠ 0) (r : ℝ) :
    o.oangle (r • o.rotation (π / 2 : ℝ) x) (r • o.rotation (π / 2 : ℝ) x - x)
      = Real.arctan r⁻¹ 
goal: o.oangle (r • (o.rotation ↑(π / 2)) x) (r • (o.rotation ↑(π / 2)) x - x) = ↑(Real.arctan r⁻¹)

full_name: Orientation.oangle_sub_left_smul_rotation_pi_div_two
code: /-- An angle in a right-angled triangle expressed using `arctan`, where one side is a multiple
of a rotation of another by `π / 2`, version subtracting vectors. -/
theorem oangle_sub_left_smul_rotation_pi_div_two {x : V} (h : x ≠ 0) (r : ℝ) :
    o.oangle (x - r • o.rotation (π / 2 : ℝ) x) x = Real.arctan r 
goal: o.oangle (x - r • (o.rotation ↑(π / 2)) x) x = ↑(Real.arctan r)

full_name: EuclideanGeometry.oangle_right_eq_arccos_of_oangle_eq_pi_div_two
code: /-- An angle in a right-angled triangle expressed using `arccos`. -/
theorem oangle_right_eq_arccos_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
    ∡ p₂ p₃ p₁ = Real.arccos (dist p₃ p₂ / dist p₁ p₃) 
goal: ∡ p₂ p₃ p₁ = ↑(Real.arccos (dist p₃ p₂ / dist p₁ p₃))

full_name: EuclideanGeometry.oangle_left_eq_arccos_of_oangle_eq_pi_div_two
code: /-- An angle in a right-angled triangle expressed using `arccos`. -/
theorem oangle_left_eq_arccos_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
    ∡ p₃ p₁ p₂ = Real.arccos (dist p₁ p₂ / dist p₁ p₃) 
goal: ∡ p₃ p₁ p₂ = ↑(Real.arccos (dist p₁ p₂ / dist p₁ p₃))

full_name: EuclideanGeometry.oangle_right_eq_arcsin_of_oangle_eq_pi_div_two
code: /-- An angle in a right-angled triangle expressed using `arcsin`. -/
theorem oangle_right_eq_arcsin_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
    ∡ p₂ p₃ p₁ = Real.arcsin (dist p₁ p₂ / dist p₁ p₃) 
goal: ∡ p₂ p₃ p₁ = ↑(Real.arcsin (dist p₁ p₂ / dist p₁ p₃))

full_name: EuclideanGeometry.oangle_left_eq_arcsin_of_oangle_eq_pi_div_two
code: /-- An angle in a right-angled triangle expressed using `arcsin`. -/
theorem oangle_left_eq_arcsin_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
    ∡ p₃ p₁ p₂ = Real.arcsin (dist p₃ p₂ / dist p₁ p₃) 
goal: ∡ p₃ p₁ p₂ = ↑(Real.arcsin (dist p₃ p₂ / dist p₁ p₃))

full_name: EuclideanGeometry.oangle_right_eq_arctan_of_oangle_eq_pi_div_two
code: /-- An angle in a right-angled triangle expressed using `arctan`. -/
theorem oangle_right_eq_arctan_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
    ∡ p₂ p₃ p₁ = Real.arctan (dist p₁ p₂ / dist p₃ p₂) 
goal: ∡ p₂ p₃ p₁ = ↑(Real.arctan (dist p₁ p₂ / dist p₃ p₂))

full_name: EuclideanGeometry.oangle_left_eq_arctan_of_oangle_eq_pi_div_two
code: /-- An angle in a right-angled triangle expressed using `arctan`. -/
theorem oangle_left_eq_arctan_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
    ∡ p₃ p₁ p₂ = Real.arctan (dist p₃ p₂ / dist p₁ p₂) 
goal: ∡ p₃ p₁ p₂ = ↑(Real.arctan (dist p₃ p₂ / dist p₁ p₂))

full_name: EuclideanGeometry.cos_oangle_right_of_oangle_eq_pi_div_two
code: /-- The cosine of an angle in a right-angled triangle as a ratio of sides. -/
theorem cos_oangle_right_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
    Real.Angle.cos (∡ p₂ p₃ p₁) = dist p₃ p₂ / dist p₁ p₃ 
goal: (∡ p₂ p₃ p₁).cos = dist p₃ p₂ / dist p₁ p₃

full_name: EuclideanGeometry.cos_oangle_left_of_oangle_eq_pi_div_two
code: /-- The cosine of an angle in a right-angled triangle as a ratio of sides. -/
theorem cos_oangle_left_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
    Real.Angle.cos (∡ p₃ p₁ p₂) = dist p₁ p₂ / dist p₁ p₃ 
goal: (∡ p₃ p₁ p₂).cos = dist p₁ p₂ / dist p₁ p₃

full_name: EuclideanGeometry.sin_oangle_right_of_oangle_eq_pi_div_two
code: /-- The sine of an angle in a right-angled triangle as a ratio of sides. -/
theorem sin_oangle_right_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
    Real.Angle.sin (∡ p₂ p₃ p₁) = dist p₁ p₂ / dist p₁ p₃ 
goal: (∡ p₂ p₃ p₁).sin = dist p₁ p₂ / dist p₁ p₃

full_name: EuclideanGeometry.sin_oangle_left_of_oangle_eq_pi_div_two
code: /-- The sine of an angle in a right-angled triangle as a ratio of sides. -/
theorem sin_oangle_left_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
    Real.Angle.sin (∡ p₃ p₁ p₂) = dist p₃ p₂ / dist p₁ p₃ 
goal: (∡ p₃ p₁ p₂).sin = dist p₃ p₂ / dist p₁ p₃

full_name: EuclideanGeometry.tan_oangle_right_of_oangle_eq_pi_div_two
code: /-- The tangent of an angle in a right-angled triangle as a ratio of sides. -/
theorem tan_oangle_right_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
    Real.Angle.tan (∡ p₂ p₃ p₁) = dist p₁ p₂ / dist p₃ p₂ 
goal: (∡ p₂ p₃ p₁).tan = dist p₁ p₂ / dist p₃ p₂

full_name: EuclideanGeometry.tan_oangle_left_of_oangle_eq_pi_div_two
code: /-- The tangent of an angle in a right-angled triangle as a ratio of sides. -/
theorem tan_oangle_left_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
    Real.Angle.tan (∡ p₃ p₁ p₂) = dist p₃ p₂ / dist p₁ p₂ 
goal: (∡ p₃ p₁ p₂).tan = dist p₃ p₂ / dist p₁ p₂

full_name: EuclideanGeometry.cos_oangle_right_mul_dist_of_oangle_eq_pi_div_two
code: /-- The cosine of an angle in a right-angled triangle multiplied by the hypotenuse equals the
adjacent side. -/
theorem cos_oangle_right_mul_dist_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P}
    (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) : Real.Angle.cos (∡ p₂ p₃ p₁) * dist p₁ p₃ = dist p₃ p₂ 
goal: (∡ p₂ p₃ p₁).cos * dist p₁ p₃ = dist p₃ p₂

full_name: EuclideanGeometry.cos_oangle_left_mul_dist_of_oangle_eq_pi_div_two
code: /-- The cosine of an angle in a right-angled triangle multiplied by the hypotenuse equals the
adjacent side. -/
theorem cos_oangle_left_mul_dist_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P}
    (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) : Real.Angle.cos (∡ p₃ p₁ p₂) * dist p₁ p₃ = dist p₁ p₂ 
goal: (∡ p₃ p₁ p₂).cos * dist p₁ p₃ = dist p₁ p₂

full_name: EuclideanGeometry.sin_oangle_right_mul_dist_of_oangle_eq_pi_div_two
code: /-- The sine of an angle in a right-angled triangle multiplied by the hypotenuse equals the
opposite side. -/
theorem sin_oangle_right_mul_dist_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P}
    (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) : Real.Angle.sin (∡ p₂ p₃ p₁) * dist p₁ p₃ = dist p₁ p₂ 
goal: (∡ p₂ p₃ p₁).sin * dist p₁ p₃ = dist p₁ p₂

full_name: EuclideanGeometry.sin_oangle_left_mul_dist_of_oangle_eq_pi_div_two
code: /-- The sine of an angle in a right-angled triangle multiplied by the hypotenuse equals the
opposite side. -/
theorem sin_oangle_left_mul_dist_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P}
    (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) : Real.Angle.sin (∡ p₃ p₁ p₂) * dist p₁ p₃ = dist p₃ p₂ 
goal: (∡ p₃ p₁ p₂).sin * dist p₁ p₃ = dist p₃ p₂

full_name: EuclideanGeometry.tan_oangle_right_mul_dist_of_oangle_eq_pi_div_two
code: /-- The tangent of an angle in a right-angled triangle multiplied by the adjacent side equals
the opposite side. -/
theorem tan_oangle_right_mul_dist_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P}
    (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) : Real.Angle.tan (∡ p₂ p₃ p₁) * dist p₃ p₂ = dist p₁ p₂ 
goal: (∡ p₂ p₃ p₁).tan * dist p₃ p₂ = dist p₁ p₂

full_name: EuclideanGeometry.tan_oangle_left_mul_dist_of_oangle_eq_pi_div_two
code: /-- The tangent of an angle in a right-angled triangle multiplied by the adjacent side equals
the opposite side. -/
theorem tan_oangle_left_mul_dist_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P}
    (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) : Real.Angle.tan (∡ p₃ p₁ p₂) * dist p₁ p₂ = dist p₃ p₂ 
goal: (∡ p₃ p₁ p₂).tan * dist p₁ p₂ = dist p₃ p₂

full_name: EuclideanGeometry.dist_div_cos_oangle_right_of_oangle_eq_pi_div_two
code: /-- A side of a right-angled triangle divided by the cosine of the adjacent angle equals the
hypotenuse. -/
theorem dist_div_cos_oangle_right_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P}
    (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) : dist p₃ p₂ / Real.Angle.cos (∡ p₂ p₃ p₁) = dist p₁ p₃ 
goal: dist p₃ p₂ / (∡ p₂ p₃ p₁).cos = dist p₁ p₃

full_name: EuclideanGeometry.dist_div_cos_oangle_left_of_oangle_eq_pi_div_two
code: /-- A side of a right-angled triangle divided by the cosine of the adjacent angle equals the
hypotenuse. -/
theorem dist_div_cos_oangle_left_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P}
    (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) : dist p₁ p₂ / Real.Angle.cos (∡ p₃ p₁ p₂) = dist p₁ p₃ 
goal: dist p₁ p₂ / (∡ p₃ p₁ p₂).cos = dist p₁ p₃

full_name: EuclideanGeometry.dist_div_sin_oangle_right_of_oangle_eq_pi_div_two
code: /-- A side of a right-angled triangle divided by the sine of the opposite angle equals the
hypotenuse. -/
theorem dist_div_sin_oangle_right_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P}
    (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) : dist p₁ p₂ / Real.Angle.sin (∡ p₂ p₃ p₁) = dist p₁ p₃ 
goal: dist p₁ p₂ / (∡ p₂ p₃ p₁).sin = dist p₁ p₃

full_name: EuclideanGeometry.dist_div_sin_oangle_left_of_oangle_eq_pi_div_two
code: /-- A side of a right-angled triangle divided by the sine of the opposite angle equals the
hypotenuse. -/
theorem dist_div_sin_oangle_left_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P}
    (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) : dist p₃ p₂ / Real.Angle.sin (∡ p₃ p₁ p₂) = dist p₁ p₃ 
goal: dist p₃ p₂ / (∡ p₃ p₁ p₂).sin = dist p₁ p₃

full_name: EuclideanGeometry.dist_div_tan_oangle_right_of_oangle_eq_pi_div_two
code: /-- A side of a right-angled triangle divided by the tangent of the opposite angle equals the
adjacent side. -/
theorem dist_div_tan_oangle_right_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P}
    (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) : dist p₁ p₂ / Real.Angle.tan (∡ p₂ p₃ p₁) = dist p₃ p₂ 
goal: dist p₁ p₂ / (∡ p₂ p₃ p₁).tan = dist p₃ p₂

full_name: EuclideanGeometry.dist_div_tan_oangle_left_of_oangle_eq_pi_div_two
code: /-- A side of a right-angled triangle divided by the tangent of the opposite angle equals the
adjacent side. -/
theorem dist_div_tan_oangle_left_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P}
    (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) : dist p₃ p₂ / Real.Angle.tan (∡ p₃ p₁ p₂) = dist p₁ p₂ 
goal: dist p₃ p₂ / (∡ p₃ p₁ p₂).tan = dist p₁ p₂