feat(geometry/euclidean/basic): lemmas about angles and distances (#7140

)





Co-authored-by: Johan Commelin <johan@commelin.net>

src/geometry/euclidean/basic.lean

@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2020 Joseph Myers. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Joseph Myers
+Authors: Joseph Myers, Manuel Candales
 -/
 import analysis.normed_space.inner_product
 import algebra.quadratic_discriminant
@@ -233,6 +233,112 @@ them is π/2. -/
 lemma inner_eq_zero_iff_angle_eq_pi_div_two (x y : V) : ⟪x, y⟫ = 0 ↔ angle x y = π / 2 :=
 iff.symm $ by simp [angle, or_imp_distrib] { contextual := tt }
 
+/-- If the angle between two vectors is π, the inner product equals the negative product
+of the norms. -/
+lemma inner_eq_neg_mul_norm_of_angle_eq_pi {x y : V} (h : angle x y = π) : ⟪x, y⟫ = - (∥x∥ * ∥y∥) :=
+by simp [← cos_angle_mul_norm_mul_norm, h]
+
+/-- If the angle between two vectors is 0, the inner product equals the product of the norms. -/
+lemma inner_eq_mul_norm_of_angle_eq_zero {x y : V} (h : angle x y = 0) : ⟪x, y⟫ = ∥x∥ * ∥y∥ :=
+by simp [← cos_angle_mul_norm_mul_norm, h]
+
+/-- The inner product of two non-zero vectors equals the negative product of their norms
+if and only if the angle between the two vectors is π. -/
+lemma inner_eq_neg_mul_norm_iff_angle_eq_pi {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
+  ⟪x, y⟫ = - (∥x∥ * ∥y∥) ↔ angle x y = π :=
+begin
+  refine ⟨λ h, _, inner_eq_neg_mul_norm_of_angle_eq_pi⟩,
+  have h₁ : (∥x∥ * ∥y∥) ≠ 0 := (mul_pos (norm_pos_iff.mpr hx) (norm_pos_iff.mpr hy)).ne',
+  rw [angle, h, neg_div, div_self h₁, real.arccos_neg_one],
+end
+
+/-- The inner product of two non-zero vectors equals the product of their norms
+if and only if the angle between the two vectors is 0. -/
+lemma inner_eq_mul_norm_iff_angle_eq_zero {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
+  ⟪x, y⟫ = ∥x∥ * ∥y∥ ↔ angle x y = 0 :=
+begin
+  refine ⟨λ h, _, inner_eq_mul_norm_of_angle_eq_zero⟩,
+  have h₁ : (∥x∥ * ∥y∥) ≠ 0 := (mul_pos (norm_pos_iff.mpr hx) (norm_pos_iff.mpr hy)).ne',
+  rw [angle, h, div_self h₁, real.arccos_one],
+end
+
+/-- If the angle between two vectors is π, the norm of their difference equals
+the sum of their norms. -/
+lemma norm_sub_eq_add_norm_of_angle_eq_pi {x y : V} (h : angle x y = π) : ∥x - y∥ = ∥x∥ + ∥y∥ :=
+begin
+  rw ← eq_of_sq_eq_sq (norm_nonneg (x - y)) (add_nonneg (norm_nonneg x) (norm_nonneg y)),
+  rw [norm_sub_pow_two_real, inner_eq_neg_mul_norm_of_angle_eq_pi h],
+  ring,
+end
+
+/-- If the angle between two vectors is 0, the norm of their sum equals
+the sum of their norms. -/
+lemma norm_add_eq_add_norm_of_angle_eq_zero {x y : V} (h : angle x y = 0) : ∥x + y∥ = ∥x∥ + ∥y∥ :=
+begin
+  rw ← eq_of_sq_eq_sq (norm_nonneg (x + y)) (add_nonneg (norm_nonneg x) (norm_nonneg y)),
+  rw [norm_add_pow_two_real, inner_eq_mul_norm_of_angle_eq_zero h],
+  ring,
+end
+
+/-- If the angle between two vectors is 0, the norm of their difference equals
+the absolute value of the difference of their norms. -/
+lemma norm_sub_eq_abs_sub_norm_of_angle_eq_zero {x y : V} (h : angle x y = 0) :
+  ∥x - y∥ = abs (∥x∥ - ∥y∥) :=
+begin
+  rw [← eq_of_sq_eq_sq (norm_nonneg (x - y)) (abs_nonneg (∥x∥ - ∥y∥)),
+      norm_sub_pow_two_real, inner_eq_mul_norm_of_angle_eq_zero h, sq_abs (∥x∥ - ∥y∥)],
+  ring,
+end
+
+/-- The norm of the difference of two non-zero vectors equals the sum of their norms
+if and only the angle between the two vectors is π. -/
+lemma norm_sub_eq_add_norm_iff_angle_eq_pi {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
+  ∥x - y∥ = ∥x∥ + ∥y∥ ↔ angle x y = π :=
+begin
+  refine ⟨λ h, _, norm_sub_eq_add_norm_of_angle_eq_pi⟩,
+  rw ← inner_eq_neg_mul_norm_iff_angle_eq_pi hx hy,
+  obtain ⟨hxy₁, hxy₂⟩ := ⟨norm_nonneg (x - y), add_nonneg (norm_nonneg x) (norm_nonneg y)⟩,
+  rw [← eq_of_sq_eq_sq hxy₁ hxy₂, norm_sub_pow_two_real] at h,
+  calc inner x y = (∥x∥ ^ 2 + ∥y∥ ^ 2 - (∥x∥ + ∥y∥) ^ 2) / 2 : by linarith
+  ...            = -(∥x∥ * ∥y∥) : by ring,
+end
+
+/-- The norm of the sum of two non-zero vectors equals the sum of their norms
+if and only the angle between the two vectors is 0. -/
+lemma norm_add_eq_add_norm_iff_angle_eq_zero {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
+  ∥x + y∥ = ∥x∥ + ∥y∥ ↔ angle x y = 0 :=
+begin
+  refine ⟨λ h, _, norm_add_eq_add_norm_of_angle_eq_zero⟩,
+  rw ← inner_eq_mul_norm_iff_angle_eq_zero hx hy,
+  obtain ⟨hxy₁, hxy₂⟩ := ⟨norm_nonneg (x + y), add_nonneg (norm_nonneg x) (norm_nonneg y)⟩,
+  rw [← eq_of_sq_eq_sq hxy₁ hxy₂, norm_add_pow_two_real] at h,
+  calc inner x y = ((∥x∥ + ∥y∥) ^ 2 - ∥x∥ ^ 2 - ∥y∥ ^ 2)/ 2 : by linarith
+  ...            = ∥x∥ * ∥y∥ : by ring,
+end
+
+/-- The norm of the difference of two non-zero vectors equals the absolute value
+of the difference of their norms if and only the angle between the two vectors is 0. -/
+lemma norm_sub_eq_abs_sub_norm_iff_angle_eq_zero {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
+  ∥x - y∥ = abs (∥x∥ - ∥y∥) ↔ angle x y = 0 :=
+begin
+  refine ⟨λ h, _, norm_sub_eq_abs_sub_norm_of_angle_eq_zero⟩,
+  rw ← inner_eq_mul_norm_iff_angle_eq_zero hx hy,
+  have h1 : ∥x - y∥ ^ 2 = (∥x∥ - ∥y∥) ^ 2, { rw h, exact sq_abs (∥x∥ - ∥y∥) },
+  rw norm_sub_pow_two_real at h1,
+  calc inner x y = ((∥x∥ + ∥y∥) ^ 2 - ∥x∥ ^ 2 - ∥y∥ ^ 2)/ 2 : by linarith
+  ...            = ∥x∥ * ∥y∥ : by ring,
+end
+
+/-- The norm of the sum of two vectors equals the norm of their difference if and only if
+the angle between them is π/2. -/
+lemma norm_add_eq_norm_sub_iff_angle_eq_pi_div_two (x y : V) :
+  ∥x + y∥ = ∥x - y∥ ↔ angle x y = π / 2 :=
+begin
+  rw [← eq_of_sq_eq_sq (norm_nonneg (x + y)) (norm_nonneg (x - y)),
+      ← inner_eq_zero_iff_angle_eq_pi_div_two x y, norm_add_pow_two_real, norm_sub_pow_two_real],
+  split; intro h; linarith,
+end
+
 end inner_product_geometry
 
 namespace euclidean_geometry
@@ -331,6 +437,89 @@ begin
   exact angle_add_angle_eq_pi_of_angle_eq_pi _ h
 end
 
+/-- Vertical Angles Theorem: angles opposite each other, formed by two intersecting straight
+lines, are equal. -/
+lemma angle_eq_angle_of_angle_eq_pi_of_angle_eq_pi {p1 p2 p3 p4 p5 : P}
+  (hapc : ∠ p1 p5 p3 = π) (hbpd : ∠ p2 p5 p4 = π) : ∠ p1 p5 p2 = ∠ p3 p5 p4 :=
+by linarith [angle_add_angle_eq_pi_of_angle_eq_pi p1 hbpd, angle_comm p4 p5 p1,
+             angle_add_angle_eq_pi_of_angle_eq_pi p4 hapc, angle_comm p4 p5 p3]
+
+/-- If ∠ABC = π then dist A B ≠ 0. -/
+lemma left_dist_ne_zero_of_angle_eq_pi {p1 p2 p3 : P} (h : ∠ p1 p2 p3 = π) : dist p1 p2 ≠ 0 :=
+begin
+  by_contra heq,
+  rw [not_not, dist_eq_zero] at heq,
+  rw [heq, angle_eq_left] at h,
+  exact real.pi_ne_zero (by linarith),
+end
+
+/-- If ∠ABC = π then dist C B ≠ 0. -/
+lemma right_dist_ne_zero_of_angle_eq_pi {p1 p2 p3 : P} (h : ∠ p1 p2 p3 = π) : dist p3 p2 ≠ 0 :=
+left_dist_ne_zero_of_angle_eq_pi $ (angle_comm _ _ _).trans h
+
+/-- If ∠ABC = π, then (dist A C) = (dist A B) + (dist B C). -/
+lemma dist_eq_add_dist_of_angle_eq_pi {p1 p2 p3 : P} (h : ∠ p1 p2 p3 = π) :
+  dist p1 p3 = dist p1 p2 + dist p3 p2 :=
+begin
+  rw [dist_eq_norm_vsub V, dist_eq_norm_vsub V, dist_eq_norm_vsub V, ← vsub_sub_vsub_cancel_right],
+  exact norm_sub_eq_add_norm_of_angle_eq_pi h,
+end
+
+/-- If A ≠ B and C ≠ B then ∠ABC = π if and only if (dist A C) = (dist A B) + (dist B C). -/
+lemma dist_eq_add_dist_iff_angle_eq_pi {p1 p2 p3 : P} (hp1p2 : p1 ≠ p2) (hp3p2 : p3 ≠ p2) :
+  dist p1 p3 = dist p1 p2 + dist p3 p2 ↔ ∠ p1 p2 p3 = π :=
+begin
+  rw [dist_eq_norm_vsub V, dist_eq_norm_vsub V, dist_eq_norm_vsub V, ← vsub_sub_vsub_cancel_right],
+  exact norm_sub_eq_add_norm_iff_angle_eq_pi
+    ((λ he, hp1p2 (vsub_eq_zero_iff_eq.1 he))) (λ he, hp3p2 (vsub_eq_zero_iff_eq.1 he)),
+end
+
+/-- If ∠ABC = 0, then (dist A C) = abs ((dist A B) - (dist B C)). -/
+lemma dist_eq_abs_sub_dist_of_angle_eq_zero {p1 p2 p3 : P} (h : ∠ p1 p2 p3 = 0) :
+  (dist p1 p3) = abs ((dist p1 p2) - (dist p3 p2)) :=
+begin
+  rw [dist_eq_norm_vsub V, dist_eq_norm_vsub V, dist_eq_norm_vsub V, ← vsub_sub_vsub_cancel_right],
+  exact norm_sub_eq_abs_sub_norm_of_angle_eq_zero h,
+end
+
+/-- If A ≠ B and C ≠ B then ∠ABC = 0 if and only if (dist A C) = abs ((dist A B) - (dist B C)). -/
+lemma dist_eq_abs_sub_dist_iff_angle_eq_zero {p1 p2 p3 : P} (hp1p2 : p1 ≠ p2) (hp3p2 : p3 ≠ p2) :
+  (dist p1 p3) = abs ((dist p1 p2) - (dist p3 p2)) ↔ ∠ p1 p2 p3 = 0 :=
+begin
+  rw [dist_eq_norm_vsub V, dist_eq_norm_vsub V, dist_eq_norm_vsub V, ← vsub_sub_vsub_cancel_right],
+  exact norm_sub_eq_abs_sub_norm_iff_angle_eq_zero
+    ((λ he, hp1p2 (vsub_eq_zero_iff_eq.1 he))) (λ he, hp3p2 (vsub_eq_zero_iff_eq.1 he)),
+end
+
+/-- The midpoint of the segment AB is the same distance from A as it is from B. -/
+lemma dist_left_midpoint_eq_dist_right_midpoint (p1 p2 : P) :
+  dist p1 (midpoint ℝ p1 p2) = dist p2 (midpoint ℝ p1 p2) :=
+by rw [dist_left_midpoint p1 p2, dist_right_midpoint p1 p2]
+
+/-- If M is the midpoint of the segment AB, then ∠AMB = π. -/
+lemma angle_midpoint_eq_pi (p1 p2 : P) (hp1p2 : p1 ≠ p2) : ∠ p1 (midpoint ℝ p1 p2) p2 = π :=
+have p2 -ᵥ midpoint ℝ p1 p2 = -(p1 -ᵥ midpoint ℝ p1 p2), by { rw neg_vsub_eq_vsub_rev, simp },
+by simp [angle, this, hp1p2]
+
+/-- If M is the midpoint of the segment AB and C is the same distance from A as it is from B
+then ∠CMA = π / 2. -/
+lemma angle_left_midpoint_eq_pi_div_two_of_dist_eq {p1 p2 p3 : P} (h : dist p3 p1 = dist p3 p2) :
+  ∠ p3 (midpoint ℝ p1 p2) p1 = π / 2 :=
+begin
+  let m : P := midpoint ℝ p1 p2,
+  have h1 : p3 -ᵥ p1 = (p3 -ᵥ m) - (p1 -ᵥ m) := (vsub_sub_vsub_cancel_right p3 p1 m).symm,
+  have h2 : p3 -ᵥ p2 = (p3 -ᵥ m) + (p1 -ᵥ m),
+  { rw [left_vsub_midpoint, ← midpoint_vsub_right, vsub_add_vsub_cancel] },
+  rw [dist_eq_norm_vsub V p3 p1, dist_eq_norm_vsub V p3 p2, h1, h2] at h,
+  exact (norm_add_eq_norm_sub_iff_angle_eq_pi_div_two (p3 -ᵥ m) (p1 -ᵥ m)).mp h.symm,
+end
+
+/-- If M is the midpoint of the segment AB and C is the same distance from A as it is from B
+then ∠CMB = π / 2. -/
+lemma angle_right_midpoint_eq_pi_div_two_of_dist_eq {p1 p2 p3 : P} (h : dist p3 p1 = dist p3 p2) :
+  ∠ p3 (midpoint ℝ p1 p2) p2 = π / 2 :=
+by rw [midpoint_comm p1 p2, angle_left_midpoint_eq_pi_div_two_of_dist_eq h.symm]
+
 /-- The inner product of two vectors given with `weighted_vsub`, in
 terms of the pairwise distances. -/
 lemma inner_weighted_vsub {ι₁ : Type*} {s₁ : finset ι₁} {w₁ : ι₁ → ℝ} (p₁ : ι₁ → P)