We study curves obtained by tracing triangle centers within special families of triangles, focusing on centers and families that yield (semi-)invariant triangle curves, meaning that varying the initial triangle changes the loci only by an affine transformation. We identify four two-parameter families of triangle centers that are semi-invariant and determine which are invariant, in the sense that the resulting curves for different initial triangles are related by a similarity transformation. We further observe that these centers, when combined with the aliquot triangle family, yield sheared Maclaurin trisectrices, whereas the nedian triangle family yields Limaçon trisectrices.

翻译：本文研究通过追踪特殊三角形族内三角形中心所获得的曲线，重点考察能够产生（半）不变三角形曲线的中心与族系，即初始三角形的变化仅通过仿射变换改变轨迹。我们识别出四个具有半不变性的双参数三角形中心族，并确定其中哪些具有不变性——即不同初始三角形生成的曲线可通过相似变换相互关联。进一步观察发现，这些中心与等分三角形族结合时会产生剪切麦克劳林三等分角线，而与中线三角形族结合则产生利马松三等分角线。