# Easy formula for coordinates of point with minimum distance to other fixed points

If you have the coordinates of 3 or more points in a 2D plane, how do you find the point that minimizes the total distance from that point to the other points? Also, is there a formula for this that can be easily memorized and used quickly in math competitions such as the AMC or AIME? I looked here (minimizing sum of distances) but I couldn't the exact formula that I was looking for.

I am aware that the point I am looking for is called the Fermat point but Wikipedia is not offering much other than a formula involving vectors, which I won't likely have the time or skill to use on competition day.

• For three points, this is known as the Fermat Point. Commented Dec 30, 2019 at 0:05
• Yes, I was aware of that. Commented Dec 30, 2019 at 0:06
• It is almost sure that no such formula exists. Even in the 3 points case, the coordinates of the so-called "Fermat point" are not given by a formula. Commented Dec 30, 2019 at 0:07
• @JeanMarie It appears to be as such. Commented Dec 30, 2019 at 0:10
• You might find the article Man versus computer, The Mathematical Gazette, Vol 91, 522, Nov 2007 of interest. This gives a coordinate system based upon the Fermat point (which therefore has position vector $<0,0,0>$). In terms of this system many results about triangles take a nice form - however, it might not be useful for a competition where the markers may well be unfamiliar with the method.
– user502266
Commented Dec 31, 2019 at 17:18