Earlier quoted context omitted.
It's certainly not always the case. In planar Euclidean geometry, for any three points, if the triangle formed by those three points has all its angles less than 120 degrees, then the sum of the distances from the three points to certain points inside the triangle will be smaller than the sum of the shortest two sides. Only if one of the angles exceeds 120 degrees will one of the three points always have the shortest…
I would expect that aircraft emissions have a fixed cost + a distance dependent variable cost. Coupled with the fact that we are optimizing over a discrete set of locations, it is very possible that the optimal location is at one of the corners of the triangle.
This all also assumes that you will always be able to fly directly to your destination on the selected day. If you end up diverting or getting delayed, or if certain routes make that _more_ likely, then this wouldn't be a win at all.