I didn’t want to put a spoiler in the question, but the area of the greatest triangle was 1344 square miles, and this is explained fully in the previous solution. The quadrilateral needs to have an area therefore of 2016 square miles (1 1/2 times 1344).

The method of achieving the greatest area with four runners is actually much more straightforward than for a triangle. The shortest and longest distances head out in diametrically opposite directions, while the other two head out in perpendicular directions.

So if the distances were, in increasing order, a b c d, the area of the quadrilateral will be (a+d)*(b+c)/2.

We know three distances: 25,33,39. The only slight issue is that we don’t know if the fourth distance will be the largest, the smallest, or somewhere in between. If we assume it lies between 25 and 39, we can say that 25 and 39 are the minimum and maximum distances, therefore:

(25+39)(33+x)/2=2016

64(33+x)=4032

(33+x)=63

x=30

And since this does indeed fall between 25 and 39, it is the correct answer.

(If we had assumed x was the maximum it would work out as x=31, which is not the maximum, and is we had assumed x was the minimum it would be 30 15/29, which is not the minimum).

The fourth runner covers 30 miles.