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Seven Region Partition (Posted on 2017-05-27) Difficulty: 3 of 5
The diameter of a polygon is defined as the longest distance between any pair of vertices of the polygon.

Partition a unit equilateral triangle into seven regions so that all seven regions have the same polygon diameter.

How small can that diameter be made?

No Solution Yet Submitted by Brian Smith    
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Some Thoughts Solution? | Comment 1 of 3
Inside the unit equilateral, draw a smaller equilateral with the same orientation and center.  Draw the three altitudes for the larger triangle, but erase them inside the smaller triangle.  This partitions the larger area into a triangle and 6 similar right parallelograms.

It is obvious to me that there is some size of interior triangle which will cause all 7 areas to have the same "polygon diameter".  I did some arithmetic, and I came up a triangle size (and a polygon diameter) of Sqrt(13) - 3, which is approximately .60555.  

Is this calculation correct?  I am not sure, but it seemed correct?

Is this the smallest possible?  I suspect so, but I have not tried to come up with other arrangements.

  Posted by Steve Herman on 2017-05-27 21:47:02
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