mathlib documentation

geometry.euclidean.angle.unoriented.right_angle

Right-angled triangles #

This file proves basic geometrical results about distances and angles in (possibly degenerate) right-angled triangles in real inner product spaces and Euclidean affine spaces.

Implementation notes #

Results in this file are generally given in a form with only those non-degeneracy conditions needed for the particular result, rather than requiring affine independence of the points of a triangle unnecessarily.

References #

Pythagorean theorem, vector angle form.

Pythagorean theorem, subtracting vectors, if-and-only-if vector angle form.

Pythagorean theorem, subtracting vectors, vector angle form.

An angle in a right-angled triangle expressed using arccos.

An angle in a right-angled triangle expressed using arcsin.

An angle in a right-angled triangle expressed using arctan.

An angle in a non-degenerate right-angled triangle is positive.

An angle in a right-angled triangle is at most π / 2.

An angle in a non-degenerate right-angled triangle is less than π / 2.

The cosine of an angle in a right-angled triangle as a ratio of sides.

The sine of an angle in a right-angled triangle as a ratio of sides.

The tangent of an angle in a right-angled triangle as a ratio of sides.

The cosine of an angle in a right-angled triangle multiplied by the hypotenuse equals the adjacent side.

The sine of an angle in a right-angled triangle multiplied by the hypotenuse equals the opposite side.

The tangent of an angle in a right-angled triangle multiplied by the adjacent side equals the opposite side.

A side of a right-angled triangle divided by the cosine of the adjacent angle equals the hypotenuse.

A side of a right-angled triangle divided by the sine of the opposite angle equals the hypotenuse.

A side of a right-angled triangle divided by the tangent of the opposite angle equals the adjacent side.

An angle in a right-angled triangle expressed using arccos, version subtracting vectors.

An angle in a right-angled triangle expressed using arcsin, version subtracting vectors.

An angle in a right-angled triangle expressed using arctan, version subtracting vectors.

An angle in a non-degenerate right-angled triangle is positive, version subtracting vectors.

An angle in a right-angled triangle is at most π / 2, version subtracting vectors.

An angle in a non-degenerate right-angled triangle is less than π / 2, version subtracting vectors.

The cosine of an angle in a right-angled triangle as a ratio of sides, version subtracting vectors.

The sine of an angle in a right-angled triangle as a ratio of sides, version subtracting vectors.

The tangent of an angle in a right-angled triangle as a ratio of sides, version subtracting vectors.

The cosine of an angle in a right-angled triangle multiplied by the hypotenuse equals the adjacent side, version subtracting vectors.

The sine of an angle in a right-angled triangle multiplied by the hypotenuse equals the opposite side, version subtracting vectors.

The tangent of an angle in a right-angled triangle multiplied by the adjacent side equals the opposite side, version subtracting vectors.

A side of a right-angled triangle divided by the cosine of the adjacent angle equals the hypotenuse, version subtracting vectors.

A side of a right-angled triangle divided by the sine of the opposite angle equals the hypotenuse, version subtracting vectors.

A side of a right-angled triangle divided by the tangent of the opposite angle equals the adjacent side, version subtracting vectors.

Pythagorean theorem, if-and-only-if angle-at-point form.

theorem euclidean_geometry.angle_eq_arccos_of_angle_eq_pi_div_two {V : Type u_1} {P : Type u_2} [inner_product_space V] [metric_space P] [normed_add_torsor V P] {p₁ p₂ p₃ : P} (h : euclidean_geometry.angle p₁ p₂ p₃ = real.pi / 2) :
euclidean_geometry.angle p₂ p₃ p₁ = real.arccos (has_dist.dist p₃ p₂ / has_dist.dist p₁ p₃)

An angle in a right-angled triangle expressed using arccos.

theorem euclidean_geometry.angle_eq_arcsin_of_angle_eq_pi_div_two {V : Type u_1} {P : Type u_2} [inner_product_space V] [metric_space P] [normed_add_torsor V P] {p₁ p₂ p₃ : P} (h : euclidean_geometry.angle p₁ p₂ p₃ = real.pi / 2) (h0 : p₁ p₂ p₃ p₂) :
euclidean_geometry.angle p₂ p₃ p₁ = real.arcsin (has_dist.dist p₁ p₂ / has_dist.dist p₁ p₃)

An angle in a right-angled triangle expressed using arcsin.

theorem euclidean_geometry.angle_eq_arctan_of_angle_eq_pi_div_two {V : Type u_1} {P : Type u_2} [inner_product_space V] [metric_space P] [normed_add_torsor V P] {p₁ p₂ p₃ : P} (h : euclidean_geometry.angle p₁ p₂ p₃ = real.pi / 2) (h0 : p₃ p₂) :
euclidean_geometry.angle p₂ p₃ p₁ = real.arctan (has_dist.dist p₁ p₂ / has_dist.dist p₃ p₂)

An angle in a right-angled triangle expressed using arctan.

theorem euclidean_geometry.angle_pos_of_angle_eq_pi_div_two {V : Type u_1} {P : Type u_2} [inner_product_space V] [metric_space P] [normed_add_torsor V P] {p₁ p₂ p₃ : P} (h : euclidean_geometry.angle p₁ p₂ p₃ = real.pi / 2) (h0 : p₁ p₂ p₃ = p₂) :
0 < euclidean_geometry.angle p₂ p₃ p₁

An angle in a non-degenerate right-angled triangle is positive.

An angle in a right-angled triangle is at most π / 2.

theorem euclidean_geometry.angle_lt_pi_div_two_of_angle_eq_pi_div_two {V : Type u_1} {P : Type u_2} [inner_product_space V] [metric_space P] [normed_add_torsor V P] {p₁ p₂ p₃ : P} (h : euclidean_geometry.angle p₁ p₂ p₃ = real.pi / 2) (h0 : p₃ p₂) :

An angle in a non-degenerate right-angled triangle is less than π / 2.

theorem euclidean_geometry.cos_angle_of_angle_eq_pi_div_two {V : Type u_1} {P : Type u_2} [inner_product_space V] [metric_space P] [normed_add_torsor V P] {p₁ p₂ p₃ : P} (h : euclidean_geometry.angle p₁ p₂ p₃ = real.pi / 2) :
real.cos (euclidean_geometry.angle p₂ p₃ p₁) = has_dist.dist p₃ p₂ / has_dist.dist p₁ p₃

The cosine of an angle in a right-angled triangle as a ratio of sides.

theorem euclidean_geometry.sin_angle_of_angle_eq_pi_div_two {V : Type u_1} {P : Type u_2} [inner_product_space V] [metric_space P] [normed_add_torsor V P] {p₁ p₂ p₃ : P} (h : euclidean_geometry.angle p₁ p₂ p₃ = real.pi / 2) (h0 : p₁ p₂ p₃ p₂) :
real.sin (euclidean_geometry.angle p₂ p₃ p₁) = has_dist.dist p₁ p₂ / has_dist.dist p₁ p₃

The sine of an angle in a right-angled triangle as a ratio of sides.

theorem euclidean_geometry.tan_angle_of_angle_eq_pi_div_two {V : Type u_1} {P : Type u_2} [inner_product_space V] [metric_space P] [normed_add_torsor V P] {p₁ p₂ p₃ : P} (h : euclidean_geometry.angle p₁ p₂ p₃ = real.pi / 2) :
real.tan (euclidean_geometry.angle p₂ p₃ p₁) = has_dist.dist p₁ p₂ / has_dist.dist p₃ p₂

The tangent of an angle in a right-angled triangle as a ratio of sides.

theorem euclidean_geometry.cos_angle_mul_dist_of_angle_eq_pi_div_two {V : Type u_1} {P : Type u_2} [inner_product_space V] [metric_space P] [normed_add_torsor V P] {p₁ p₂ p₃ : P} (h : euclidean_geometry.angle p₁ p₂ p₃ = real.pi / 2) :
real.cos (euclidean_geometry.angle p₂ p₃ p₁) * has_dist.dist p₁ p₃ = has_dist.dist p₃ p₂

The cosine of an angle in a right-angled triangle multiplied by the hypotenuse equals the adjacent side.

theorem euclidean_geometry.sin_angle_mul_dist_of_angle_eq_pi_div_two {V : Type u_1} {P : Type u_2} [inner_product_space V] [metric_space P] [normed_add_torsor V P] {p₁ p₂ p₃ : P} (h : euclidean_geometry.angle p₁ p₂ p₃ = real.pi / 2) :
real.sin (euclidean_geometry.angle p₂ p₃ p₁) * has_dist.dist p₁ p₃ = has_dist.dist p₁ p₂

The sine of an angle in a right-angled triangle multiplied by the hypotenuse equals the opposite side.

theorem euclidean_geometry.tan_angle_mul_dist_of_angle_eq_pi_div_two {V : Type u_1} {P : Type u_2} [inner_product_space V] [metric_space P] [normed_add_torsor V P] {p₁ p₂ p₃ : P} (h : euclidean_geometry.angle p₁ p₂ p₃ = real.pi / 2) (h0 : p₁ = p₂ p₃ p₂) :
real.tan (euclidean_geometry.angle p₂ p₃ p₁) * has_dist.dist p₃ p₂ = has_dist.dist p₁ p₂

The tangent of an angle in a right-angled triangle multiplied by the adjacent side equals the opposite side.

theorem euclidean_geometry.dist_div_cos_angle_of_angle_eq_pi_div_two {V : Type u_1} {P : Type u_2} [inner_product_space V] [metric_space P] [normed_add_torsor V P] {p₁ p₂ p₃ : P} (h : euclidean_geometry.angle p₁ p₂ p₃ = real.pi / 2) (h0 : p₁ = p₂ p₃ p₂) :
has_dist.dist p₃ p₂ / real.cos (euclidean_geometry.angle p₂ p₃ p₁) = has_dist.dist p₁ p₃

A side of a right-angled triangle divided by the cosine of the adjacent angle equals the hypotenuse.

theorem euclidean_geometry.dist_div_sin_angle_of_angle_eq_pi_div_two {V : Type u_1} {P : Type u_2} [inner_product_space V] [metric_space P] [normed_add_torsor V P] {p₁ p₂ p₃ : P} (h : euclidean_geometry.angle p₁ p₂ p₃ = real.pi / 2) (h0 : p₁ p₂ p₃ = p₂) :
has_dist.dist p₁ p₂ / real.sin (euclidean_geometry.angle p₂ p₃ p₁) = has_dist.dist p₁ p₃

A side of a right-angled triangle divided by the sine of the opposite angle equals the hypotenuse.

theorem euclidean_geometry.dist_div_tan_angle_of_angle_eq_pi_div_two {V : Type u_1} {P : Type u_2} [inner_product_space V] [metric_space P] [normed_add_torsor V P] {p₁ p₂ p₃ : P} (h : euclidean_geometry.angle p₁ p₂ p₃ = real.pi / 2) (h0 : p₁ p₂ p₃ = p₂) :
has_dist.dist p₁ p₂ / real.tan (euclidean_geometry.angle p₂ p₃ p₁) = has_dist.dist p₃ p₂

A side of a right-angled triangle divided by the tangent of the opposite angle equals the adjacent side.