Consider the operation of insetting every side of a polygon by a fixed perpendicular distance, as illustrated (for example) at https://alienryderflex.com/polygon_inset/. For which polygons does this operation produce a similar polygon (for at least some interval of inset distances)?
For example, it does for all regular polygons, all triangles, and all kites (and hence rhombuses) -- intuitively because an instance of each of these families is determined up to similarity by its vertex angles, which are preserved by the insetting operation. But it definitely does not for all polygons -- not the example in the link above, nor for any non-square rectangles. Is there a characterization of polygons for which insets are similar?
My brother completed the proof, which he said he preferred I post.
Claim: A polygon is inset-similar (even for just one inset) iff it is tangential (i.e., has an inscribed circle).
Proof: It is pretty straightforward that a tangential polygon is inset-similar for any inset less than its inradius: scale by an appropriate factor about the incenter, and all the new edges are the same distance from the original ones.
Now, given a polygon which is inset-similar by an inset $h_1$, let the ratio of similarity of the inset polygon to the original be $r<1$. Then scaling down the diagram of the original and inset polygon by factor of $r$, we see there is an inset $h_2$ at which the original polygon is also inset-similar, but this time with a scale factor of $r^2$. Repeating, we get a sequence of inset polygons at insets $h_i$ at scale factor $r^i$. Now notice that the vertices of each of these inset polygons lie on the angle bisectors of the original vertex angles. Since $r^i \to 0$, these angle bisectors must all meet at a point. Hence, by a well-known theorem (see https://en.wikipedia.org/wiki/Tangential_polygon) the original polygon is tangential. (And it turns out its inradius is the limit of the $h_i$.)