dist_div_cos_oangle_center_eq_two_mul_radius

English

The sum of twice a base angle and twice a tangent-based apex angle equals π, for triangles inscribed in a sphere.

Русский

Сумма двухразовых углов основания и вершины касательной окружности равна π.

LaTeX

$$2 oangle(p3 p1 s.center) + 2 oangle(p1 p2 p3) = π$$

Lean4

/-- Given two points on a circle, twice the radius of that circle may be expressed explicitly as
the distance between those two points divided by the cosine of the angle between the chord and
the radius at one of those points. -/
theorem dist_div_cos_oangle_center_eq_two_mul_radius {s : Sphere P} {p₁ p₂ : P} (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s)
    (h : p₁ ≠ p₂) : dist p₁ p₂ / Real.Angle.cos (∡ p₂ p₁ s.center) = 2 * s.radius := by
  rw [← dist_div_cos_oangle_center_div_two_eq_radius hp₁ hp₂ h, mul_div_cancel₀ _ (two_ne_zero' ℝ)]

Похожие утверждения