There's a four point Fermat problem that can't be solved by linearizing the function for the sum of the distances of the unknown point.
If one tries one ends up with division by zero. The gradient of L at point c is not continuous.
The distance function near a point is cone shaped.
The individual gradients are not well behaved near a given point as this plot shows.
For a problem like this one can compute the gradient function for two points displaced from the minimum and try to find where two lines in their directions through the chosen points intersect to get a better estimate of the minimum.
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