I find the “solution region” questions interesting as well, but think I would save that for our exploration of systems of linear inequalities (whereas I would use this question in my linear systems unit).

All that to say, I think both of your suggestions would certainly make for interesting alternative explorations.

My students and I encountered something similar when we searched for a location for a meeting place or a distribution center that was equal distance from several people or stores. Pairing several points to “construct” perpendicular bisectors usually produced a region from which to choose a optimal location. Simplifying the search to just three locations (points), represented by the vertices of a triangle, produces a single intersection point for the perpendicular bisectors (as long as the original points are not collinear).

GeoGebra is a nice tool for creating a dynamic representation to investigate under what conditions the intersections of the perpendicular bisector lines will coincide.

http://www.achieve.org/files/CCSS-CTE-Rabbit-Food-FINAL.pdf

There needs to be some modification for my kids but it’s an awesome task. The issue boils down to “optimization” of the triply shaded area, which will be one of the points of intersection.

Not sure how to explain the the optimal point must be a point of intersection. Would love some help on this one…