tan_angle_of_angle_eq_pi_div_two — Mathlib

English

If ∠ p1 p2 p3 = π/2, then dist(p1,p2) = tan(∠ p2 p3 p1) · dist(p3,p2).

Русский

Если ∠ p1 p2 p3 = π/2, тогда dist(p1,p2) = tan(∠ p2 p3 p1) · dist(p3,p2).

LaTeX

$$$ \operatorname{dist}(p_1,p_2) = \tan(\angle p_2 p_3 p_1) \cdot \operatorname{dist}(p_3,p_2) $$$

Lean4

/-- 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 :=
  by
  rw [angle, ← inner_eq_zero_iff_angle_eq_pi_div_two, real_inner_comm, ← neg_eq_zero, ← inner_neg_left,
    neg_vsub_eq_vsub_rev] at h
  rw [angle, dist_eq_norm_vsub V p₁ p₂, dist_eq_norm_vsub' V p₃ p₂, ← vsub_add_vsub_cancel p₁ p₂ p₃, add_comm,
    tan_angle_add_of_inner_eq_zero h]

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