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 Seven Region Partition (Posted on 2017-05-27)
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 No Rating

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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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