This study was motivated by the well-known Euler triangle formula. We extended the concept of inscribed triangles with the same incenter to those with the same centroid and the same orthocenter. The existence and constructibility of such centroid-sharing and orthocenter-sharing triangles were elegantly characterized using the nine-point circle. Focusing further on the envelope of the sides of centroid-sharing inscribed triangles, we first identified the corresponding focus and then characterized the envelope as an ellipse, a hyperbola, or determined the conditions under which it degenerated. Notably, by unifying the cases of triangles with the same incenter, orthocenter, and centroid, we discovered a geometric relationship in which the areas formed a geometric progression. Secondly, we considered circumscribed triangles with the same center. This construction proved significantly more challenging, and we employed the Nagel line as a key tool in the construction process. Finally, we investigated both the envelope of the sides of centroid-sharing inscribed triangles and the locus of vertices of centroid-sharing circumscribed triangles. For both cases, we provided geometric interpretations of the major and minor axes, as well as the focal distance for the ellipse, and the transverse and conjugate axes and focal distance for the hyperbola.