I'm trying to find a way to calculate the point at which the total distance from other points is at a minimum. In other words I'm trying to find a way to find the point at which the function $$d(x,y,z)=\sqrt{(x-x_1)^2+(y-y_1)^2}+\sqrt{(x-x_2)^2+(y-y_2)^2}+\sqrt{(x-x_3)^2+(y-y_3)^2}$$ is at an absolute minimum. Does such a solution exist?
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@DavidK . Also known as the Steiner point. – DanielWainfleet Oct 20 '21 at 01:37
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@DanielWainfleet I hadn't heard that one. https://mathworld.wolfram.com/SteinerPoints.html – David K Oct 20 '21 at 02:12
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@DavidK . Steiner investigated the Q of connecting a finite number $n\ge 3$ of planar points with a path of minimal total length. This is called the Steiner Point Problem. – DanielWainfleet Oct 20 '21 at 06:18
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@DanielWainfleet Got it. The Fermat Point occurs in the solution for the Steiner tree for $n=3.$ Nice generalization. – David K Oct 20 '21 at 13:09
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@DavidK . There is a variant called the Orthogonal Steiner Point Problem where the connecting line segments must be parallel to one or the other co-ordinate axes. I have heard this affects the layout of connections of components within an integrated circuit. – DanielWainfleet Oct 20 '21 at 14:06