oangle_add_oangle_add_oangle_eq_pi

English

Converse of pons asinorum for oriented angles: sum of angles equals π.

Русский

Обратное утверждение оропорц-понс-асино: сумма углов равна π.

LaTeX

$$$\angle x y p + \angle y p x = \angle x p y$$$

Lean4

/-- The **sum of the angles of a triangle** (possibly degenerate, where the triangle is a line),
oriented angles at point. -/
theorem oangle_add_oangle_add_oangle_eq_pi [Module.Oriented ℝ V (Fin 2)] [Fact (Module.finrank ℝ V = 2)] {p₁ p₂ p₃ : P}
    (h21 : p₂ ≠ p₁) (h32 : p₃ ≠ p₂) (h13 : p₁ ≠ p₃) : ∡ p₁ p₂ p₃ + ∡ p₂ p₃ p₁ + ∡ p₃ p₁ p₂ = π := by
  simpa only [neg_vsub_eq_vsub_rev] using
    positiveOrientation.oangle_add_cyc3_neg_left (vsub_ne_zero.mpr h21) (vsub_ne_zero.mpr h32) (vsub_ne_zero.mpr h13)

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