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Points On A Circle (Posted on 2004-08-14) Difficulty: 4 of 5
I placed six points on the circumference of a circle such that the distance between any two of the points is an integer. What is the smallest such circle I could use?

What if each distance must be unique?

No Solution Yet Submitted by ThoughtProvoker    
Rating: 4.4286 (7 votes)

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Solution with non-distinct sides | Comment 11 of 17 |

When the sides are allowed to be non-distinct, there is a solution with radius 7/sqrt(3) = 4.01415...

the hexagon has side lengths 3,5,3,5,3,5, and diagonals of length 7 and 8. Labelling the vertices A-F, the distances are as follows:

AB,CD,EF: 3

BC,DE,FA: 5

AC,BD,CE,DF,EA,FB: 7

AD,BE,CF: 8

working on the distinct sides version. initial results suggest the radius > 500. For a quadrilateral with distinct chord lengths, the smallest radius is 12.8102523...., and for a pentagon the smallest radius is 39.2938971....


  Posted by sundberg on 2004-08-19 01:29:47
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