The classification of convex polygons with triangular area or perimeter bisecting deltoids

We classify all convex polygons whose area-bisecting deltoids or perimeter-bisecting deltoids are similar to those for a triangle, that is, they are tri-cusped and tri-concave-out closed curves. The additional condition that these two kinds of deltoids are segment-free makes no difference to the first classification and restricts the second to one that is much more similar to the first. We show that, up to similarity, the restricted second class is a complete system of representatives for the first class modulo affine equivalence.