Discovered by William Chapple in 1746, the Poristic family is a set of variable-perimeter triangles with common Incircle and Circumcircle. By definition, the family has constant Inradius-to-Circumradius ratio. Interestingly, this invariance also holds for the family of 3-periodics in the Elliptic Billiard, though here Inradius and Circumradius are variable and perimeters are constant. Indeed, we show one family is mapped onto the other via a varying similarity transform. This implies that any scale-free quantities and invariants observed in one family must hold on the other.

中文翻译：

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与相似性相关：椭圆形台球中的多孔三角形和3周期

由威廉·查普尔（William Chapple）于1746年发现，Poristic家族是一组可变周长的三角形，它们具有常见的圆和圆。根据定义，该族的辐射半径与圆周半径之比是恒定的。有趣的是，这种不变性也适用于椭圆台球的3周期族，尽管此处Inradius和Circumradius是变量，周长是恒定的。实际上，我们显示了一个家庭通过变化的相似性变换映射到了另一个家庭。这意味着在一个家庭中观察到的任何无标度的数量和不变式都必须保留在另一个家庭中。