AI 中文总结
本文将经典七圆定理推广至任意闭合六圆链，通过构造相切圆证明相关圆心连线共点，经典定理为其退化情形。
AI 中文摘要
七圆定理指出，若六个圆构成闭合链且与一个公共圆相切，则公共圆上相对切点的三条连线共点。我们将该结果推广至任意闭合六圆链：给定此类链，考虑两个圆，每个圆分别与链中交替的三个圆之一相切；对链中的每个圆，构造一个与它及上述两个圆都相切的圆。我们证明，链中相对成员对应的新构造圆的圆心的三条连线共点。在经典构型中，与交替三元组关联的两个圆重合为公共圆，六个新构造圆退化为六个切点（视为半径为零的圆），因此经典七圆定理可作为我们推广结果的退化情形被导出。
英文摘要
The Seven Circles Theorem states that if six circles form a closed chain and are tangent to a common circle, then the three lines joining opposite points of tangency on the common circle are concurrent. We extend this result to an arbitrary closed six-circle chain. Given such a chain, we consider two circles, each tangent to one of the two alternating triples of circles in the chain. For each circle in the chain, we then construct a circle tangent to it and to both of these circles. We prove that the three lines joining the centers of the newly constructed circles corresponding to opposite members of the chain are concurrent. In the classical configuration, the two circles associated with the alternating triples coincide with the common circle, while the six newly constructed circles degenerate to the six points of tangency, viewed as circles of radius zero. Thus, the classical Seven Circles Theorem is recovered as a degenerate case of our generalization.
