dist_div_tan_oangle_left_of_oangle_eq_pi_div_two

English

If ∡ p1 p2 p3 = π/2, then dist(p1,p2) / dist(p1,p3) = cos(∡ p3 p1 p2).

Русский

Если ∡ p1 p2 p3 = π/2, то dist(p1,p2) / dist(p1,p3) = cos(∡ p3 p1 p2).

LaTeX

$$$\cos\left(\angle p_3 p_1 p_2\right) = \frac{\operatorname{dist}(p_1,p_2)}{\operatorname{dist}(p_1,p_3)} \quad\text{при } \angle p_1 p_2 p_3 = \frac{\pi}{2}$$$

Lean4

/-- 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 :=
  by
  have hs : (∡ p₃ p₁ p₂).sign = 1 := by rw [← oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]
  rw [oangle_eq_angle_of_sign_eq_one hs, angle_comm, Real.Angle.tan_coe,
    dist_div_tan_angle_of_angle_eq_pi_div_two (angle_rev_eq_pi_div_two_of_oangle_eq_pi_div_two h)
      (Or.inl (right_ne_of_oangle_eq_pi_div_two h))]

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