Curve equidistant from x lines and y points.

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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?