Create new feature point at minimum distance between numerous points

[CaseyCalamaio]

Is there a tool or workflow that can create a feature at minimum distance or equal distance between several points? This is a general question but to clarify I've attached a couple images to show what I mean with three points:

Below are a couple candidate points:

Here I've created a some Thiessen Polygons:

Then I convert the vertices of the Thiessen polygons to points and select the point at the center:

This is just a simple example of what I'd like to learn how to do on a larger scale, e.g. to create more than one "minimum distance" points over an area containing a lot more candidate points. The goal is to automate this as well so the user doesn't have to go in and select the points located between the desired points and omit those on the edge.

Any thoughts, advice, or references to similar posts would be greatly appreciated. Thanks!

Case

[JoshuaBixby]

If you haven't read them already, decent discussions with some good links:  Algorithm to find point of minimum total distance from locations and Geometric median of multidimensional points

[CaseyCalamaio]

Thanks for the reference

[JoshuaBixby]

Forgot to add the obvious:  Median Center.

[CaseyCalamaio]

Excellent. Thanks for the link. This might be exactly what I was looking for.

[DanPatterson_Retired]

The triangle centroid is then what you want Centroid - Wikipedia  your only problem will be to form the triplets necessary for its calculation, if you are using existing points.

[CaseyCalamaio]

Thank you very much, this helps a lot and makes good sense. I will see what I can do with these calculations and let you all know what results.