angle_lt_pi_div_two_of_angle_eq_pi_div_two

English

If ∠ p1 p2 p3 = π/2 and p3 ≠ p2, then ∠ p2 p3 p1 < π/2.

Русский

Если ∠ p1 p2 p3 = π/2 и p3 ≠ p2, то ∠ p2 p3 p1 < π/2.

LaTeX

$$$ \angle p_2 p_3 p_1 < \frac{\pi}{2} $$$

Lean4

/-- An angle in a non-degenerate right-angled triangle is less than `π / 2`. -/
theorem angle_lt_pi_div_two_of_angle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2) (h0 : p₃ ≠ p₂) :
    ∠ p₂ p₃ p₁ < π / 2 :=
  by
  rw [angle, ← inner_eq_zero_iff_angle_eq_pi_div_two, real_inner_comm, ← neg_eq_zero, ← inner_neg_left,
    neg_vsub_eq_vsub_rev] at h
  rw [ne_comm, ← @vsub_ne_zero V] at h0
  rw [angle, ← vsub_add_vsub_cancel p₁ p₂ p₃, add_comm]
  exact angle_add_lt_pi_div_two_of_inner_eq_zero h h0

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