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Given a triangle in a plane with vertices A, B, C, find the point T that minimizes the sum of distances between A-T, B-T, and C-T.

I can experimentally determine this point by sampling the space and finding the minimum via an attractive, interactive web page: http://phrogz.net/SVG/tweedlie-gradient.xhtml

Drag any vertex to see the gradient of value sums change and the point T be updated. I find it interesting to note that moving any vertex towards or away from T does not affect T, until the point where the vertex touches T and causes it to move along with it.

However, an experimental determination is neither efficient nor accurate, and prevents me from graphing other interesting properties such as the arcs of the vertices when a fixed sum is required.

Please help me find the equation for T (preferably in terms of vectors, as the answer should be the same in 2D or 3D).

Edit: Based on the answer, I've created an updated interactive using the calculated point and showing the construction: http://phrogz.net/SVG/fermat-point.xhtml

Screenshot of visualization

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+1 for the link –  amWhy Jun 9 '11 at 18:40
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1 Answer

up vote 2 down vote accepted

Take a look at http://en.wikipedia.org/wiki/Fermat_point

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Very helpful, thanks! I could follow through with the math on my own, but it would be helpful for Stack Overflow posterity if you chose to include a full solution in your answer instead of just linking to a (possibly-but-probably-not ephemeral) 3rd party website. –  Phrogz Jun 9 '11 at 18:27
    
My thinking is like this: if someone else has done the work before of explaining this, then it is not much use for me to explain all the things again. –  Beni Bogosel Jun 9 '11 at 18:33
    
Sorry, to be clear, I was not suggesting that you quote or rewrite the entire Wikipedia article. Rather, the article you cite describes properties of the point and how to geometrically construct the answer, but does not in fact include an equation (or two, for the >120° case) that directly answers this question. I was suggesting that your answer would become more valuable than the Wikipedia article (to a certain set of people) if you included the equation. –  Phrogz Jun 9 '11 at 18:35
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