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Given points $(x_1,y_1),(x_2,y_2),...(x_n,y_n)$ and lines of the form $y=a_1x+b_1, y=a_2x+b_2,...y=a_nx+b_n$ what is the curve such that the sum of the distance from all points and lines is equal to some constant? What are these curves called?

tox123
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    For general points and lines, the curve would probably be called "nonexistent" or "empty". Consider, for instance, a tic-tac-toe board with unit spacing; if the constant is less than 2, then the set of solutions is empty. – John Hughes Aug 29 '15 at 15:40
  • If we agree to consider the sum of squares of the distances then the problem is workable. Otherwise one might be tempted to resort to Lagrange's multipliers. – Adelafif Aug 29 '15 at 15:45
  • @Adelafif what is that? – tox123 Aug 29 '15 at 21:15

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