发表机构
The University of Texas at Arlington（德克萨斯大学阿灵顿分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文推广Dan Reznik的几何观察结果，基于Marden定理的三角形对称参数化给出更强形式的证明，还推广了其奇数边形相关结论，并提出圆内接n边形外切于中心二次曲线时的正方形总面积不变性新猜想。
AI 中文摘要
本文将Dan Reznik在一系列计算机辅助实验中得出的若干近期观察结果进行了推广。受其中一项原始观察结果近期获证的启发，我们通过将其与涉及中线的经典度量恒等式关联，给出了该观察结果更强形式的简短证明。我们的证明基于由Marden定理直接得到的三角形对称参数化，该方法不依赖于内接二次曲线必须是包含在外接圆内的椭圆这一假设。所得结果同样适用于焦点位于外接圆外的内接椭圆以及内接双曲线。我们还证明了Reznik关于内接于一对位似椭圆且外切于该对椭圆的奇数边形的第二项观察结果的自然推广，该证明同样适用于交叉奇数边形。最后，我们提出了一个新猜想：关于外切于中心二次曲线的圆内接n边形各边上所作正方形总面积的不变性，从而将这些面积不变现象纳入统一的几何框架中。
英文摘要
We prove that the total area of the power circles associated with an odd $p$-gon inscribed in and circumscribed about a pair of homothetic ellipses remains invariant throughout the Poncelet family. Our proof is based on a simple affine averaging principle, which also yields several related quadratic invariants, including explicit formulas for the sums of the squared distances from the center to the vertices, to the side midpoints, as well as for the sum of the squared side lengths. As an application, we show that a convex Poncelet pentagon and the corresponding star Poncelet pentagon, both circumscribed about the same inellipse, have equal total power-circle area. These results unify several metric invariants of odd Poncelet polygons within a common affine-geometric framework.
Comments9 pages, 4 figures
