English
Twice the oriented angles being equal implies a relation equating collinearity patterns in two triples.
Русский
Удвоенные ориентированные углы равны, значит структуры коллинеарности в двух троях совпадают.
LaTeX
$$$ (2 : \mathbb{Z}) \!\cdot \angle p_1 p_2 p_3 = (2 : \mathbb{Z}) \!\cdot \angle p_4 p_5 p_6 \Rightarrow (Collinear Real \{p_1,p_2,p_3\}) \iff (Collinear Real \{p_4,p_5,p_6\})$$$
Lean4
/-- Given three points not equal to `p`, the angle between the first and the third at `p` minus
the angle between the second and the third equals the angle between the first and the second. -/
@[simp]
theorem oangle_sub_right {p p₁ p₂ p₃ : P} (hp₁ : p₁ ≠ p) (hp₂ : p₂ ≠ p) (hp₃ : p₃ ≠ p) :
∡ p₁ p p₃ - ∡ p₂ p p₃ = ∡ p₁ p p₂ :=
o.oangle_sub_right (vsub_ne_zero.2 hp₁) (vsub_ne_zero.2 hp₂) (vsub_ne_zero.2 hp₃)
