We study center power with respect to circles derived from Poncelet 3-periodics (triangles) in a generic pair of ellipses as well as loci of their triangle centers. We show that (i) for any concentric pair, the power of the center with respect to either circumcircle or Euler's circle is invariant, and (ii) if a triangle center of a 3-periodic in a generic nested pair is a fixed linear combination of barycenter and circumcenter, its locus over the family is an ellipse.

中文翻译：

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Poncelet三角形的中心功率和轨迹

我们研究在一般椭圆形对中从庞塞特3周期（三角形）派生的圆以及其三角形中心的轨迹的中心功率。我们证明（i）对于任何同心对，中心相对于外接圆或欧拉圆的幂是不变的；并且（ii）如果一般嵌套对中3周期的三角形中心是固定的线性组合属于重心和外心，在家庭中的位置是椭圆形。