"Four Is Not Enough" and distance between two sets

Question

I have recently read Patrick Honner's article Four is Not Enough in Quanta magazine. I have been thinking about the bonus question you can find at the end of the worksheet. I think I have found the length of the white segment in the picture, which is $\frac{\sqrt{7}}{2}d$, but I have a doubt: how I can show that this is the minimum distance between any two points in the hexagons? Can anyone give me a hint, please?

Answer 1

Draw two perpendiculars at the endpoints of the white segment.

Answer 2

A hint as required:

Use the white segment as an $x$-axis, and any perpendicular as a $y$-axis. Show (for instance via the angles between the white segment and the sides of the hexagons) that each extremity of the white segment corresponds to the point with the largest (respectively, lowest) $x$-value over the whole corresponding hexagon.