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

full_name: InnerProductGeometry.norm_add_sq_eq_norm_sq_add_norm_sq_iff_angle_eq_pi_div_two
code: /-- Pythagorean theorem, if-and-only-if vector angle form. -/
theorem norm_add_sq_eq_norm_sq_add_norm_sq_iff_angle_eq_pi_div_two (x y : V) :
    ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ ↔ angle x y = π / 2 
goal: ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ ↔ InnerProductGeometry.angle x y = π / 2

full_name: InnerProductGeometry.norm_add_sq_eq_norm_sq_add_norm_sq'
code: /-- Pythagorean theorem, vector angle form. -/
theorem norm_add_sq_eq_norm_sq_add_norm_sq' (x y : V) (h : angle x y = π / 2) :
    ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ 
goal: ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖

full_name: InnerProductGeometry.norm_sub_sq_eq_norm_sq_add_norm_sq_iff_angle_eq_pi_div_two
code: /-- Pythagorean theorem, subtracting vectors, if-and-only-if vector angle form. -/
theorem norm_sub_sq_eq_norm_sq_add_norm_sq_iff_angle_eq_pi_div_two (x y : V) :
    ‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ ↔ angle x y = π / 2 
goal: ‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ ↔ InnerProductGeometry.angle x y = π / 2

full_name: InnerProductGeometry.norm_sub_sq_eq_norm_sq_add_norm_sq'
code: /-- Pythagorean theorem, subtracting vectors, vector angle form. -/
theorem norm_sub_sq_eq_norm_sq_add_norm_sq' (x y : V) (h : angle x y = π / 2) :
    ‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ 
goal: ‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖

full_name: InnerProductGeometry.angle_add_eq_arccos_of_inner_eq_zero
code: /-- An angle in a right-angled triangle expressed using `arccos`. -/
theorem angle_add_eq_arccos_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
    angle x (x + y) = Real.arccos (‖x‖ / ‖x + y‖) 
goal: InnerProductGeometry.angle x (x + y) = Real.arccos (‖x‖ / ‖x + y‖)

full_name: InnerProductGeometry.angle_add_eq_arcsin_of_inner_eq_zero
code: /-- An angle in a right-angled triangle expressed using `arcsin`. -/
theorem angle_add_eq_arcsin_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x ≠ 0 ∨ y ≠ 0) :
    angle x (x + y) = Real.arcsin (‖y‖ / ‖x + y‖) 
goal: InnerProductGeometry.angle x (x + y) = Real.arcsin (‖y‖ / ‖x + y‖)

full_name: InnerProductGeometry.angle_add_eq_arctan_of_inner_eq_zero
code: /-- An angle in a right-angled triangle expressed using `arctan`. -/
theorem angle_add_eq_arctan_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x ≠ 0) :
    angle x (x + y) = Real.arctan (‖y‖ / ‖x‖) 
goal: InnerProductGeometry.angle x (x + y) = Real.arctan (‖y‖ / ‖x‖)

full_name: InnerProductGeometry.angle_add_pos_of_inner_eq_zero
code: /-- An angle in a non-degenerate right-angled triangle is positive. -/
theorem angle_add_pos_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x = 0 ∨ y ≠ 0) :
    0 < angle x (x + y) 
goal: 0 < InnerProductGeometry.angle x (x + y)

full_name: InnerProductGeometry.angle_add_le_pi_div_two_of_inner_eq_zero
code: /-- An angle in a right-angled triangle is at most `π / 2`. -/
theorem angle_add_le_pi_div_two_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
    angle x (x + y) ≤ π / 2 
goal: InnerProductGeometry.angle x (x + y) ≤ π / 2

full_name: InnerProductGeometry.angle_add_lt_pi_div_two_of_inner_eq_zero
code: /-- An angle in a non-degenerate right-angled triangle is less than `π / 2`. -/
theorem angle_add_lt_pi_div_two_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x ≠ 0) :
    angle x (x + y) < π / 2 
goal: InnerProductGeometry.angle x (x + y) < π / 2

full_name: InnerProductGeometry.cos_angle_add_of_inner_eq_zero
code: /-- The cosine of an angle in a right-angled triangle as a ratio of sides. -/
theorem cos_angle_add_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
    Real.cos (angle x (x + y)) = ‖x‖ / ‖x + y‖ 
goal: Real.cos (InnerProductGeometry.angle x (x + y)) = ‖x‖ / ‖x + y‖

full_name: InnerProductGeometry.sin_angle_add_of_inner_eq_zero
code: /-- The sine of an angle in a right-angled triangle as a ratio of sides. -/
theorem sin_angle_add_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x ≠ 0 ∨ y ≠ 0) :
    Real.sin (angle x (x + y)) = ‖y‖ / ‖x + y‖ 
goal: Real.sin (InnerProductGeometry.angle x (x + y)) = ‖y‖ / ‖x + y‖

full_name: InnerProductGeometry.tan_angle_add_of_inner_eq_zero
code: /-- The tangent of an angle in a right-angled triangle as a ratio of sides. -/
theorem tan_angle_add_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
    Real.tan (angle x (x + y)) = ‖y‖ / ‖x‖ 
goal: Real.tan (InnerProductGeometry.angle x (x + y)) = ‖y‖ / ‖x‖

full_name: InnerProductGeometry.cos_angle_add_mul_norm_of_inner_eq_zero
code: /-- The cosine of an angle in a right-angled triangle multiplied by the hypotenuse equals the
adjacent side. -/
theorem cos_angle_add_mul_norm_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
    Real.cos (angle x (x + y)) * ‖x + y‖ = ‖x‖ 
goal: Real.cos (InnerProductGeometry.angle x (x + y)) * ‖x + y‖ = ‖x‖

full_name: InnerProductGeometry.sin_angle_add_mul_norm_of_inner_eq_zero
code: /-- The sine of an angle in a right-angled triangle multiplied by the hypotenuse equals the
opposite side. -/
theorem sin_angle_add_mul_norm_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
    Real.sin (angle x (x + y)) * ‖x + y‖ = ‖y‖ 
goal: Real.sin (InnerProductGeometry.angle x (x + y)) * ‖x + y‖ = ‖y‖

full_name: InnerProductGeometry.tan_angle_add_mul_norm_of_inner_eq_zero
code: /-- The tangent of an angle in a right-angled triangle multiplied by the adjacent side equals
the opposite side. -/
theorem tan_angle_add_mul_norm_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x ≠ 0 ∨ y = 0) :
    Real.tan (angle x (x + y)) * ‖x‖ = ‖y‖ 
goal: Real.tan (InnerProductGeometry.angle x (x + y)) * ‖x‖ = ‖y‖

full_name: InnerProductGeometry.norm_div_cos_angle_add_of_inner_eq_zero
code: /-- A side of a right-angled triangle divided by the cosine of the adjacent angle equals the
hypotenuse. -/
theorem norm_div_cos_angle_add_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x ≠ 0 ∨ y = 0) :
    ‖x‖ / Real.cos (angle x (x + y)) = ‖x + y‖ 
goal: ‖x‖ / Real.cos (InnerProductGeometry.angle x (x + y)) = ‖x + y‖

full_name: InnerProductGeometry.norm_div_sin_angle_add_of_inner_eq_zero
code: /-- A side of a right-angled triangle divided by the sine of the opposite angle equals the
hypotenuse. -/
theorem norm_div_sin_angle_add_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x = 0 ∨ y ≠ 0) :
    ‖y‖ / Real.sin (angle x (x + y)) = ‖x + y‖ 
goal: ‖y‖ / Real.sin (InnerProductGeometry.angle x (x + y)) = ‖x + y‖

full_name: InnerProductGeometry.norm_div_tan_angle_add_of_inner_eq_zero
code: /-- A side of a right-angled triangle divided by the tangent of the opposite angle equals the
adjacent side. -/
theorem norm_div_tan_angle_add_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x = 0 ∨ y ≠ 0) :
    ‖y‖ / Real.tan (angle x (x + y)) = ‖x‖ 
goal: ‖y‖ / Real.tan (InnerProductGeometry.angle x (x + y)) = ‖x‖

full_name: InnerProductGeometry.angle_sub_eq_arccos_of_inner_eq_zero
code: /-- An angle in a right-angled triangle expressed using `arccos`, version subtracting vectors. -/
theorem angle_sub_eq_arccos_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
    angle x (x - y) = Real.arccos (‖x‖ / ‖x - y‖) 
goal: InnerProductGeometry.angle x (x - y) = Real.arccos (‖x‖ / ‖x - y‖)

full_name: InnerProductGeometry.angle_sub_eq_arcsin_of_inner_eq_zero
code: /-- An angle in a right-angled triangle expressed using `arcsin`, version subtracting vectors. -/
theorem angle_sub_eq_arcsin_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x ≠ 0 ∨ y ≠ 0) :
    angle x (x - y) = Real.arcsin (‖y‖ / ‖x - y‖) 
goal: InnerProductGeometry.angle x (x - y) = Real.arcsin (‖y‖ / ‖x - y‖)

full_name: InnerProductGeometry.angle_sub_eq_arctan_of_inner_eq_zero
code: /-- An angle in a right-angled triangle expressed using `arctan`, version subtracting vectors. -/
theorem angle_sub_eq_arctan_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x ≠ 0) :
    angle x (x - y) = Real.arctan (‖y‖ / ‖x‖) 
goal: InnerProductGeometry.angle x (x - y) = Real.arctan (‖y‖ / ‖x‖)

full_name: InnerProductGeometry.angle_sub_pos_of_inner_eq_zero
code: /-- An angle in a non-degenerate right-angled triangle is positive, version subtracting
vectors. -/
theorem angle_sub_pos_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x = 0 ∨ y ≠ 0) :
    0 < angle x (x - y) 
goal: 0 < InnerProductGeometry.angle x (x - y)

full_name: InnerProductGeometry.angle_sub_le_pi_div_two_of_inner_eq_zero
code: /-- An angle in a right-angled triangle is at most `π / 2`, version subtracting vectors. -/
theorem angle_sub_le_pi_div_two_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
    angle x (x - y) ≤ π / 2 
goal: InnerProductGeometry.angle x (x - y) ≤ π / 2

full_name: InnerProductGeometry.angle_sub_lt_pi_div_two_of_inner_eq_zero
code: /-- An angle in a non-degenerate right-angled triangle is less than `π / 2`, version subtracting
vectors. -/
theorem angle_sub_lt_pi_div_two_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x ≠ 0) :
    angle x (x - y) < π / 2 
goal: InnerProductGeometry.angle x (x - y) < π / 2

full_name: InnerProductGeometry.cos_angle_sub_of_inner_eq_zero
code: /-- The cosine of an angle in a right-angled triangle as a ratio of sides, version subtracting
vectors. -/
theorem cos_angle_sub_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
    Real.cos (angle x (x - y)) = ‖x‖ / ‖x - y‖ 
goal: Real.cos (InnerProductGeometry.angle x (x - y)) = ‖x‖ / ‖x - y‖

full_name: InnerProductGeometry.sin_angle_sub_of_inner_eq_zero
code: /-- The sine of an angle in a right-angled triangle as a ratio of sides, version subtracting
vectors. -/
theorem sin_angle_sub_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x ≠ 0 ∨ y ≠ 0) :
    Real.sin (angle x (x - y)) = ‖y‖ / ‖x - y‖ 
goal: Real.sin (InnerProductGeometry.angle x (x - y)) = ‖y‖ / ‖x - y‖

full_name: InnerProductGeometry.tan_angle_sub_of_inner_eq_zero
code: /-- The tangent of an angle in a right-angled triangle as a ratio of sides, version subtracting
vectors. -/
theorem tan_angle_sub_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
    Real.tan (angle x (x - y)) = ‖y‖ / ‖x‖ 
goal: Real.tan (InnerProductGeometry.angle x (x - y)) = ‖y‖ / ‖x‖

full_name: InnerProductGeometry.cos_angle_sub_mul_norm_of_inner_eq_zero
code: /-- The cosine of an angle in a right-angled triangle multiplied by the hypotenuse equals the
adjacent side, version subtracting vectors. -/
theorem cos_angle_sub_mul_norm_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
    Real.cos (angle x (x - y)) * ‖x - y‖ = ‖x‖ 
goal: Real.cos (InnerProductGeometry.angle x (x - y)) * ‖x - y‖ = ‖x‖

full_name: InnerProductGeometry.sin_angle_sub_mul_norm_of_inner_eq_zero
code: /-- The sine of an angle in a right-angled triangle multiplied by the hypotenuse equals the
opposite side, version subtracting vectors. -/
theorem sin_angle_sub_mul_norm_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
    Real.sin (angle x (x - y)) * ‖x - y‖ = ‖y‖ 
goal: Real.sin (InnerProductGeometry.angle x (x - y)) * ‖x - y‖ = ‖y‖

full_name: InnerProductGeometry.tan_angle_sub_mul_norm_of_inner_eq_zero
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_angle_sub_mul_norm_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x ≠ 0 ∨ y = 0) :
    Real.tan (angle x (x - y)) * ‖x‖ = ‖y‖ 
goal: Real.tan (InnerProductGeometry.angle x (x - y)) * ‖x‖ = ‖y‖

full_name: InnerProductGeometry.norm_div_cos_angle_sub_of_inner_eq_zero
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_angle_sub_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x ≠ 0 ∨ y = 0) :
    ‖x‖ / Real.cos (angle x (x - y)) = ‖x - y‖ 
goal: ‖x‖ / Real.cos (InnerProductGeometry.angle x (x - y)) = ‖x - y‖

full_name: InnerProductGeometry.norm_div_sin_angle_sub_of_inner_eq_zero
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_angle_sub_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x = 0 ∨ y ≠ 0) :
    ‖y‖ / Real.sin (angle x (x - y)) = ‖x - y‖ 
goal: ‖y‖ / Real.sin (InnerProductGeometry.angle x (x - y)) = ‖x - y‖

full_name: InnerProductGeometry.norm_div_tan_angle_sub_of_inner_eq_zero
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_angle_sub_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x = 0 ∨ y ≠ 0) :
    ‖y‖ / Real.tan (angle x (x - y)) = ‖x‖ 
goal: ‖y‖ / Real.tan (InnerProductGeometry.angle x (x - y)) = ‖x‖

full_name: EuclideanGeometry.dist_sq_eq_dist_sq_add_dist_sq_iff_angle_eq_pi_div_two
code: /-- **Pythagorean theorem**, if-and-only-if angle-at-point form. -/
theorem dist_sq_eq_dist_sq_add_dist_sq_iff_angle_eq_pi_div_two (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 
goal: dist p1 p3 * dist p1 p3 = dist p1 p2 * dist p1 p2 + dist p3 p2 * dist p3 p2 ↔ ∠ p1 p2 p3 = π / 2

full_name: EuclideanGeometry.angle_eq_arccos_of_angle_eq_pi_div_two
code: /-- An angle in a right-angled triangle expressed using `arccos`. -/
theorem angle_eq_arccos_of_angle_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.angle_eq_arcsin_of_angle_eq_pi_div_two
code: /-- An angle in a right-angled triangle expressed using `arcsin`. -/
theorem angle_eq_arcsin_of_angle_eq_pi_div_two {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) 
goal: ∠ p₂ p₃ p₁ = Real.arcsin (dist p₁ p₂ / dist p₁ p₃)

full_name: EuclideanGeometry.angle_eq_arctan_of_angle_eq_pi_div_two
code: /-- An angle in a right-angled triangle expressed using `arctan`. -/
theorem angle_eq_arctan_of_angle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2)
    (h0 : p₃ ≠ p₂) : ∠ 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.angle_pos_of_angle_eq_pi_div_two
code: /-- An angle in a non-degenerate right-angled triangle is positive. -/
theorem angle_pos_of_angle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2)
    (h0 : p₁ ≠ p₂ ∨ p₃ = p₂) : 0 < ∠ p₂ p₃ p₁ 
goal: 0 < ∠ p₂ p₃ p₁

full_name: EuclideanGeometry.angle_le_pi_div_two_of_angle_eq_pi_div_two
code: /-- An angle in a right-angled triangle is at most `π / 2`. -/
theorem angle_le_pi_div_two_of_angle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2) :
    ∠ p₂ p₃ p₁ ≤ π / 2 
goal: ∠ p₂ p₃ p₁ ≤ π / 2

full_name: EuclideanGeometry.angle_lt_pi_div_two_of_angle_eq_pi_div_two
code: /-- An angle in a non-degenerate right-angled triangle is less than `π / 2`. -/
theorem angle_lt_pi_div_two_of_angle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2)
    (h0 : p₃ ≠ p₂) : ∠ p₂ p₃ p₁ < π / 2 
goal: ∠ p₂ p₃ p₁ < π / 2

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

full_name: EuclideanGeometry.sin_angle_of_angle_eq_pi_div_two
code: /-- The sine of an angle in a right-angled triangle as a ratio of sides. -/
theorem sin_angle_of_angle_eq_pi_div_two {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 
goal: Real.sin (∠ p₂ p₃ p₁) = dist p₁ p₂ / dist p₁ p₃

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

full_name: EuclideanGeometry.cos_angle_mul_dist_of_angle_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_angle_mul_dist_of_angle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2) :
    Real.cos (∠ p₂ p₃ p₁) * dist p₁ p₃ = dist p₃ p₂ 
goal: Real.cos (∠ p₂ p₃ p₁) * dist p₁ p₃ = dist p₃ p₂

full_name: EuclideanGeometry.sin_angle_mul_dist_of_angle_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_angle_mul_dist_of_angle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2) :
    Real.sin (∠ p₂ p₃ p₁) * dist p₁ p₃ = dist p₁ p₂ 
goal: Real.sin (∠ p₂ p₃ p₁) * dist p₁ p₃ = dist p₁ p₂

full_name: EuclideanGeometry.tan_angle_mul_dist_of_angle_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_angle_mul_dist_of_angle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2)
    (h0 : p₁ = p₂ ∨ p₃ ≠ p₂) : Real.tan (∠ p₂ p₃ p₁) * dist p₃ p₂ = dist p₁ p₂ 
goal: Real.tan (∠ p₂ p₃ p₁) * dist p₃ p₂ = dist p₁ p₂

full_name: EuclideanGeometry.dist_div_cos_angle_of_angle_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_angle_of_angle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2)
    (h0 : p₁ = p₂ ∨ p₃ ≠ p₂) : dist p₃ p₂ / Real.cos (∠ p₂ p₃ p₁) = dist p₁ p₃ 
goal: dist p₃ p₂ / Real.cos (∠ p₂ p₃ p₁) = dist p₁ p₃

full_name: EuclideanGeometry.dist_div_sin_angle_of_angle_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_angle_of_angle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2)
    (h0 : p₁ ≠ p₂ ∨ p₃ = p₂) : dist p₁ p₂ / Real.sin (∠ p₂ p₃ p₁) = dist p₁ p₃ 
goal: dist p₁ p₂ / Real.sin (∠ p₂ p₃ p₁) = dist p₁ p₃

full_name: EuclideanGeometry.dist_div_tan_angle_of_angle_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_angle_of_angle_eq_pi_div_two {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 
goal: dist p₁ p₂ / Real.tan (∠ p₂ p₃ p₁) = dist p₃ p₂