Tato Red Envelope for Chinese New Year

Geometry Level 4

Chinese New Year is about to come soon. Michael did not have any red envelope in his house, so he started with a smooth origami paper. He created the Tato envelope, which resembles a unit square as shown above, where $$B, D, F$$ and $$H$$ are the midpoints of $$\overline{AC}$$, $$\overline{CE}$$, $$\overline{EG}$$ and $$\overline{GA}$$, respectively. Each of the lines within the square is projected from the given points.

What is the area of the patterned region in the diagram? If your answer can be expressed as $$\dfrac{m}{n}$$, where $$m$$ and $$n$$ are coprime positive integers, evaluate $$m + n$$.

Fun Hint: Figure out how to create the Tato envelope as shown above. Before reaching the final steps to do some computation, carefully observe the angles of the polygons. This should help you answer the problem.

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