Octagon divided into 6 parts
A father died and left his 6 sons an octagon field with 8 equal side lengths but different angles, as shown in the figure below.
The 6 sons want to divide the field up into 6 congruent parts, but do not care about the position or orientation.
Using only a ruler, can we divide the field into 6 identical parts?
Source: French Mathematical Olympiad 1988 .
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1 solution
For the rare few who were lucky in guessing the correct answer but are not certain how it's dissected into 6 congruent parts, below is the graphic showing how it's done. The relation to the pentagon is obvious. It's possible to accurately sketch out the dissection entirely with the use of just a straightedge, given the perimeter.
Below is the same graphic with green construction lines included. One can see the two pentagons side-by-side, each pentagon being composed of 4 acute isosceles triangles and 3 smaller obtuse isosceles triangles. Together with 4 more acute isosceles triangles at the top and bottom, there is a total of 12 acute isosceles triangles and 6 smaller obtuse isosceles triangles, 1/6 of which are 2 acute isosceles triangles and 1 smaller obtuse isosceles triangle. That suggests that one of the pieces might be composed of just that.