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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.

Soham Konar
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    For three points, this is known as the Fermat Point. – ViHdzP Dec 30 '19 at 00:05
  • Yes, I was aware of that. – Soham Konar Dec 30 '19 at 00:06
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    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. – Jean Marie Dec 30 '19 at 00:07
  • @JeanMarie It appears to be as such. – Soham Konar Dec 30 '19 at 00:10
  • The problem is rather easy to solve for any number of points but the explcit formula does not exist. If you are interested by the method, let me know. – Claude Leibovici Dec 30 '19 at 06:19
  • @ClaudeLeibovici Please do let me know. – Soham Konar Dec 30 '19 at 06:48
  • Have a look at my answer at https://math.stackexchange.com/questions/3453660/find-the-x-y-location-of-one-point-unknown-location-in-space-with-respect-to-f/3454103#3454103 . If it is not clear, just tell. Cheers :-) – Claude Leibovici Dec 30 '19 at 06:55
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    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. –  Dec 31 '19 at 17:18
  • @ClaudeLeibovici I understand your answer thank you so much! – Soham Konar Jan 01 '20 at 16:53

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