English
The sign of the signedInfDist equals the sign of the corresponding excenter weight.
Русский
Знак подписи расстояния совпадает с соглавной весовой функцией эксцентра.
LaTeX
$$SignType.sign(s.signedInfDist i (s.excenter signs)) = SignType.sign (s.excenterWeights signs i)$$
Lean4
theorem sign_signedInfDist_excenter {signs : Finset (Fin (n + 1))} (h : s.ExcenterExists signs) (i : Fin (n + 1)) :
SignType.sign (s.signedInfDist i (s.excenter signs)) = SignType.sign (s.excenterWeights signs i) :=
by
rw [excenter_eq_affineCombination, signedInfDist_affineCombination _ _ h.sum_excenterWeights_eq_one, sign_mul]
convert mul_one _
rw [sign_eq_one_iff, ← dist_eq_norm_vsub]
exact s.height_pos _