Encyclopedia of Triangle Centers
faculty.evansville.edu
faculty.evansville.edu
Even for the 3 points case the solution is not 100% trivial.
Edit: Ah. I get it now. If the points made an equilateral triangle you'd want to put the disc on the triangle center. That sounds like a fun problem..
For a larger number of points it's generally the same, only you don't know which of the multiple possible triangles will work.
It's about 20k lines worth of fixed position text, pretty easy for a modern computer.
I wonder what the pixel 3 browser is doing differently, isn't it also Chrome? Pixel 3 surely has a better CPU than computers in the year 2000, and even then you could scroll through 20k text lines easily.
I worked for the IT department at the University of Evansville for many years, leaving just three years ago. I've had more than one run-in with this particular site. This site is maintained by hand (no auto-gen) by Dr. Kimberling. He's an extremely smart individual (I've had many classes with him), and he prefers to keep things simple.
Until about 2 year prior to me leaving he kept a secondary computer in his office that was only for running QBasic (an old AMD-K6 running Windows 2K, off the network). He finally gave that up and transitioned to using Mathematica.
The obvious counterpart for this (I think) would be to have a set of "standard" triangles, including a couple with random vertex positions that hopefully satisfy no "interesting" mathematical relationships, and report (say) 12 decimal places of the coordinates of each centre for each of those triangles. Then, if you run across some mysterious point in a triangle, you can look for its coordinates and see whether they match any of the centres.
(This would also give a way of verifying that no two of the thousands of triangle centres catalogued here are coincidentally equivalent to one another.)
The On-Line Encyclopedia of Integer Sequences