In plane geometry, we often see that the locus of a particular geometric object remains invariant in all possible configurations of a set of points. For example, let $ABC$ be a triangle and $X,Y,Z$ be points on lines $BC,CA,AB$. Let $\omega_1,\omega_2,\omega_3$ be three circles with diameters $\overline{AX},\overline{BY},\overline{CZ}$, respectively. Then, the locus of the radical center of $\omega_1,\omega_2$, and $\omega_3$ is independent of the choice of points $X, Y, Z$: It is always the orthocenter of the triangle $ABC$. (This proposition may be proved using the fact that the reflections of the orthocenter through the sides lie on the circumcircle.)
I would be glad if anyone could provide other examples of such geometric structures.