A simple geometry question that fools almost everyone

Why? You're still measuring the distance between the two points on the line. One is in the middle and one is not.

You could instead fold up the lines connected to the D point, into a V shape with the D point at the bottom, such that the points B and C (the top tips of the V) are separated horizontally by length 2 (and vertically by 0), and use the length 5 and 3 sides to go 4 units straight left or right from the top center of the V, and place point A and connect back to the far side of the V. Now point A is 3.87298 units above and 4.0 units to the side of point D, a diagonal length of 5.56776.

You could continuously deform this shape back down to the straight line case.

Now that I think about it, in the process of that deformation, there might be a point where slightly widening the V would allow you to place the point slightly higher and get an even larger max distance.