norm_div_tan_oangle_add_left_of_oangle_eq_pi_div_two

English

If o.oangle x y = π/2, then o.oangle y (y−x) = arctan( ||x|| / ||y|| ).

Русский

Если o.oangle x y = π/2, то o.oangle y (y−x) = arctan( ||x|| / ||y|| ).

LaTeX

$$$ o.oangle x y = \uparrow\left( \frac{\pi}{2} \right) \Rightarrow o.oangle y (y-x) = \operatorname{Real.Angle.coe}\left( \operatorname{Real.arctan}\left( \frac{\|x\|}{\|y\|} \right) \right) $$$

Lean4

/-- A side of a right-angled triangle divided by the tangent of the opposite angle equals the
adjacent side. -/
theorem norm_div_tan_oangle_add_left_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    ‖x‖ / Real.Angle.tan (o.oangle (x + y) y) = ‖y‖ :=
  by
  rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
  rw [add_comm]
  exact (-o).norm_div_tan_oangle_add_right_of_oangle_eq_pi_div_two h

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