Puzzle of the Week #587 - Greatest Quadrilateral

Last time, three friends each ran 25, 33 and 39 miles from the same staring point to maximise the area of the triangle formed by their endpoints.

This time, those same friends want to enlist a fourth friend. The three original runners will run the same distances as before. If they want the area of the resulting quadrilateral to be one and a half times that of the triangle from last time, what is the minimum distance must the fourth runner cover?

 

NB: if you didn’t solve the previous puzzle or don’t want to, please see the solution to that puzzle.