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Given 3 points on a plane, all 3 at the same semi-plane defined by a line (e), find a point P on the line (e) for which the sum of lengths of the 3 segments that are defined (by each of the 3 points and the point on the line), is minimal.

I think that we must draw the projections of each of the 3 initial points A, B, C, say, A', B' and C' and then take the middle point M of A'B', then the middle point N of MC' and point N is the one we ask - or something like this. But even if it is correct, I don't know how to prove it :( Any ideas are much appreciated!

Tom Galle
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  • I doubt there's anything as simple as your proposed construct. See the referenced papers for what some people call the n-ellipse. Your setup is essentially finding the tangency of the 3-ellipse contour with a straight line "outside" of the triangle formed by the 3 given points. By looking at the relevant equations, one can technically translate any exponentiation and square roots etc to geometric constructs. I would very much like to see the final result turning out to be "nice", but it's highly unlikely. – Lee David Chung Lin Mar 23 '18 at 14:13
  • One obvious way to see that your construction is wrong: the result should not depend on the labeling of the points, but if the three projected points are all distinct then your construction gives a different result for each choice of which of the three points to label $C.$ Actually, it is easy to construct two cases that have the exact same projected points but different solutions, so there cannot be any correct solution that relies on just the three projected points. – David K Aug 26 '18 at 16:43

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