oangle_sub_left — Mathlib

English

If dist p1 p2 equals dist p1 p3, then ∡ p1 p2 p3 equals ∡ p2 p3 p1 (cyclic shift of the triple).

Русский

Если расстояния от p1 до p2 и p1 до p3 равны, то ∡ p1 p2 p3 равно ∡ p2 p3 p1.

LaTeX

$$$ dist(p_1,p_2) = dist(p_1,p_3) \Rightarrow oangle p_1 p_2 p_3 = oangle p_2 p_3 p_1$$$

Lean4

/-- Given three points not equal to `p`, the angle between the first and the third at `p` minus
the angle between the first and the second equals the angle between the second and the third. -/
@[simp]
theorem oangle_sub_left {p p₁ p₂ p₃ : P} (hp₁ : p₁ ≠ p) (hp₂ : p₂ ≠ p) (hp₃ : p₃ ≠ p) :
    ∡ p₁ p p₃ - ∡ p₁ p p₂ = ∡ p₂ p p₃ :=
  o.oangle_sub_left (vsub_ne_zero.2 hp₁) (vsub_ne_zero.2 hp₂) (vsub_ne_zero.2 hp₃)

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