We provide a simple proof of the existence of a planar separator by showing that it is an easy consequence of the circle packing theorem. We also reprove other results on separators, including: (A) There is a simple cycle separator if the planar graph is triangulated. Furthermore, if each face has at most $d$ edges on its boundary, then there is a cycle separator of size O(sqrt{d n}). (B) For a set of n balls in R^d, that are k-ply, there is a separator, in the intersection graph of the balls, of size O(k^{1/d}n^{1-1/d}). (C) The k nearest neighbor graph of a set of n points in R^d contains a separator of size O(k^{1/d} n^{1-1/d}). The new proofs are (arguably) significantly simpler than previous proofs.

翻译：我们通过展示平面分隔子的存在性是圆填充定理的一个简单推论，给出了其存在性的一个简单证明。我们还重新证明了关于分隔子的其他结果，包括：(A) 若平面图是三角化的，则存在简单环分隔子。此外，若每个面的边界上至多有 $d$ 条边，则存在大小为 O(sqrt{d n}) 的环分隔子。(B) 对于 R^d 空间中 k-层叠的 n 个球，在球的交图中存在一个大小为 O(k^{1/d}n^{1-1/d}) 的分隔子。(C) R^d 空间中 n 个点集的 k 最近邻图包含一个大小为 O(k^{1/d} n^{1-1/d}) 的分隔子。这些新证明（可以说）比以往的证明显著更简单。