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I have a project in 10th grade math. we are meant to find a point that is the closest distance to 3 points in a triangle. The purpose is to build a railway station that is closest to 3 landmarks. We have taken a brief look on special segments of triangles in class. (orthocenter, centroid, incenter, and circumcenter). How can I find the point closest to the three points in a triangle? Thank you

Excuse my basic knowledge, I think what I mean by "closest" is the shortest average distance from the "railway station" to each of the "landmarks". The purpose of the project is to find the ideal position for the "railway" station. I'd appreciate it if you gave the best options you know of with a brief description that I can then research further. :)

Here is an image http://i.stack.imgur.com/ye5fD.jpg

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    $\begingroup$ You have to define what you mean by "closest distance to 3 points" first. Smallest maximum distance? Smallest sum of distances? Smallest sum of squares of distances? $\endgroup$ – Gerry Myerson Sep 19 '15 at 10:26
  • $\begingroup$ Gerry Myerson is right, what is meant by closest? May be You are looking for the center of the circumscribed circle, its the point of intersection of the perpendicular bisectors. This has the same distance to all three of the points, namely the radius of the circumsribed circle $\endgroup$ – Peter Melech Sep 19 '15 at 10:33
  • $\begingroup$ a point cannot be closest to the three vertices simultaneously unless the distances are same. I think this will be the minimum sum of squares of distances or else something like this. $\endgroup$ – user249332 Sep 19 '15 at 10:39
  • $\begingroup$ It generally helps to provide some drawing, even if not neat. $\endgroup$ – NoChance Sep 19 '15 at 10:43
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    $\begingroup$ Does this help? en.m.wikipedia.org/wiki/Fermat_point $\endgroup$ – Michael Hoppe Sep 19 '15 at 11:50
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I think you are looking for Fermat point. Read this wiki page for more details https://en.wikipedia.org/wiki/Fermat_point

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    $\begingroup$ It would be better to cite the essentials and then post the link $\endgroup$ – Shailesh Dec 23 '15 at 7:18

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