But it occurred to me it's easy to show the sum of squared distances to diagonally opposite corners are the same.

Let horizontal and vertical lines through the cow divide the fences into lengths u and v vertically and w and x horizontally.

Squared distance to one corner is u

^{2}+ x

^{2}and to the diagonally opposite corner, v

^{2}+ w

^{2}.

Squared distances to the other two corners are u

^{2}+ w

^{2}and v

^{2}+ x

^{2}.

The squared distances in both cases add to u

^{2}+ x

^{2}+ w

^{2}+ v

^{2}.