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Is there a way to construct a convex hexagon (need not be regular) such that the number of rational "distances" in it is maximized. A distance is defined as the segment joining any two points in a hexagon. Hexagon has 15 such distances $({}_6 \mathrm{ C }_2)$.

Further, what is the maximum number of such distances which will be integers ?

For example, in a regular hexagon, there are 3 types of distances which can be made from one point to the others, as shown below.

p, q and r are the distances

These distances are respectively 1,sqrt(3) and 2. Is it possible to make a hexagon where maximum such triplets will be rational ?

Eeshan
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  • Use "distance," not "dimension." – David G. Stork Aug 19 '22 at 06:02
  • It is possible to have a hexagon where all the distances between pairs of points are integers. The question has come up here before, maybe a search for integral hexagon will find it. – Gerry Myerson Aug 19 '22 at 07:13
  • See https://math.stackexchange.com/questions/3647840/how-to-construct-six-points-abcdef-on-a-plane-so-that-the-distance-between-any and my comments there. Also, https://math.stackexchange.com/questions/237291/points-placing-in-plane – Gerry Myerson Aug 19 '22 at 07:19
  • Thanks @GerryMyerson – Eeshan Aug 20 '22 at 06:24

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