# Find the point in a triangle minimizing the sum of distances to the vertices

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

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