Previously we showed the family of 3-periodics in the elliptic billiard (confocal pair) is the image under a variable similarity transform of poristic triangles (those with non-concentric, fixed incircle and circumcircle). Both families conserve the ratio of inradius to circumradius and therefore also the sum of cosines. This is consisten with the fact that a similarity preserves angles. Here we study two new Poncelet 3-periodic families also tied to each other via a variable similarity: (i) a first one interscribed in a pair of concentric, homothetic ellipses, and (ii) a second non-concentric one known as the Brocard porism: fixed circumcircle and Brocard inellipse. The Brocard points of this family are stationary at the foci of the inellipse. A key common invariant is the Brocard angle, and therefore the sum of cotangents. This raises an interesting question: given a non-concentric Poncelet family (limited or not to the outer conic being a circle), can a similar doppelg\"anger always be found interscribed in a concentric, axis-aligned ellipse and/or conic pair?

中文翻译：

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相似性 II 相关：Poncelet 3-Periodics in the Homothetic Porism 和 Brocard Porism

之前我们展示的椭圆台球（共焦对）中的三周期族是多孔三角形（具有非同心、固定内圆和外接圆）的可变相似变换下的图像。两个家族都守恒内半径与圆周半径的比率，因此也守恒余弦之和。这与相似性保留角度的事实一致。在这里，我们研究了两个新的 Poncelet 3 周期族，它们也通过变量相似性相互联系：（i）第一个内切于一对同心的同位椭圆中，以及（ii）第二个非同心椭圆，称为 Brocard棱镜：固定外接圆和布罗卡椭圆。该族的 Brocard 点在椭圆的焦点上是静止的。一个关键的共同不变量是布罗卡角，因此是余切之和。