Caption: A diagram illustrating the centers of mass of 2-dimensional objects (i.e., 2-dimensional systems) of uniform density with high symmetry.
One can find these centers of mass by inspection.
Features:
The points of highest symmetry are the points about which each object can be rotated into itself by any angle which is multiple of a unit angle specific to the object: e.g., infinitesimal for the circle, 90° for the square, 72° for the regular pentagon and pentagram, etc.
Now consider a distinct second reflection axis. By the same argument as above, the center of mass must lie on this second reflection axis.
Since there is only one center of mass, the center of mass must be at the intersection of the 2 distinct reflection axes. QED.
This mental trick does NOT work for the regular pentagon and pentagram. However, if you think of symmetrical sets of 5 points ...