Maximal minimum distance between points in a rectangle of size $17\times 32$

Question

I am training in problem solving and this one greatly resists to my understanding:

Given a rectangle with specified length and width, you have to place 5 points in the rectangle such that you maximise the distance between the five points. I have to return the highest possible minimum distance between any two points.

If the figure is a square or a rectangle whose length is much greater than its width I can easily figure out the answer. However if the rectangle is of size lets say $17\times 32$, I can't figure out the correct answer. This works me up… I am not asking for the answer, merely a few hints?

Any input is appreciated !

Thanks

Answer 1

This is similar to circle packing, where the centers of the circles are the points you want to place in the rectangle. Have a look at this link (second table where $n=4$). Notice how the center circle moves down the more the rectangle deviates from a square.

Using this as inspiration, for your $17 \times 32$ example the second configuration given for $n = 4$ is the way to go. In this configuration the centers of the circles make two equilateral triangles and the max distance $d$ is about $d = 17.19$.