Locus of a Point where the
Sum of Distances from Four Fixed Points
is a Constant

Square (0,0), (0,1), (1,1), (1,0)

In this example, the four fixed points are the vertices of a unit square. The constants inside toward the outside range from 2.84 to 4.0.

Given that the vertices are on a unit square then the minimum sum would be .




Repeating, the same sketch with a different graphing application, the values of  K are from outside toward the center are  4.0,  3.75, 3.50, 3.25, 3.0, and 2.84







Trapezoid (-1,0), (0,1), (1,1), (2,0

The constant terms must be larger because the minimum sum of the distances is going to be greater than 4.








Here is the same problem using a different graphing application:  The values of K from outside inward are 7.0, 6.75, 6.50, 6.25, 6.00, 5.75, 5.00, 4.75, and 4.5.











Further Explorations -

1. Try for points at the vertices of other quadrilaterals.

2. Try for points at the vertices of a non-convex quadrilateral.

3. Try for more that 4 points.