sum_mongePointWeightsWithCircumcenter

English

The preimage of a sphere under inversion with center c and radius R is the union of the center c with the perpendicular bisector of c and inv(c,R)y, for a sphere with center y and radius dist(y,c).

Русский

Предобраз сферы под инверсией с центром c и радиусом R равен объединению точки c и перп биссектором c и inv(c,R)y для сферы с центром y и радиусом dist(y,c).

LaTeX

$$$$ \operatorname{Set.preimage}(\mathrm{inversion}(c,R), \mathrm{sphere}(y, \operatorname{dist}(y,c))) = \{c\} \cup (\perpBisector(c, \mathrm{inversion}(c,R)\,y)). $$$$

Lean4

/-- `mongePointWeightsWithCircumcenter` sums to 1. -/
@[simp]
theorem sum_mongePointWeightsWithCircumcenter (n : ℕ) : ∑ i, mongePointWeightsWithCircumcenter n i = 1 :=
  by
  simp_rw [sum_pointsWithCircumcenter, mongePointWeightsWithCircumcenter, sum_const, card_fin, nsmul_eq_mul]
  simp [field]
  ring

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