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| Seven Region Partition (Posted on 2017-05-27) |
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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?
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Submitted by Brian Smith
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Solution:
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This arrangement has diameter D = (sqrt(3) - 1)/2 ~= 0.366
Draw the three altitudes for each side.
Draw three circles of radius D, one centered at each triangle vertex. Each circle will intersect two sides and one altitude.
For each circle, draw the two segments joining the point on the altitude to the two points on the sides. This forms 3 kites.
Finally, draw the three segments from the three interior points. This forms 3 trapezoids and an equilateral triangle in the center.
Each kite has two sides and both diagonals equal to D. The central triangle has sides equal to D. Each trapezoid has its longer base and both diagonals equal to D.
Jer provides an alternate explanation here. |
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