tan_angle_mul_dist_of_angle_eq_pi_div_two

English

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

Русский

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

LaTeX

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

Lean4

/-- 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 :=
  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 [ne_comm, ← @vsub_ne_zero V, ← @vsub_eq_zero_iff_eq V, or_comm] at h0
  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_mul_norm_of_inner_eq_zero h h0]

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