Finding points equidistant from the corners of a convex polygon in plane

Question

Let $Z_1, Z_2,......,Z_n$ be the no corners of convex polygon in the argand plane. We may assume $Z_k=(x_k,y_k), k=1,2,3,..,n$ .Is it possible to find the points inside the polygon which are equidistant from the corners? I remain obliged for any help .

Answer 1

In general no. Such a point would be the circumcentre of every triangle with vertices among the $Z_j$s. In general these circumcentres won't be the same. There are special cases (rectangles, regular polygons etc.) where they will be the same, though.

Answer 2

Take the circumcenter $C$ of the triangle $\triangle Z_1Z_2Z_3$. It either is equidistant from all corners or it isn't. If it is, then the set of all points equidistant from all corners is $\{C\}$; otherwise, it is the empty set.