Incredibly hard geometry problem from Russia

This challenging problem was given to students in Russia, but it stumped me! The question was presented as text only--no diagram. Also, the problem was part of an entrance exam for a selective high school. Can you solve it? From each corner of a parallelogram, draw two line segments to connect the corner to the midpoints of the opposite sides. These eight line segments bound an octagon. What is the ratio of the octagon’s area to the parallelogram’s area?

Thanks to Michael K. for suggesting the problem and sending the solution!

My blog post for this video (also links to many other solutions):
https://wp.me/p6aMk-5k7

Here's another solution write-up using symmetry by Bob Corns:
https://docs.google.com/document/d/1t...

Here's a solution from Reuven BarYehuda which is very nicely explained:
   • The Octagon In A Parallelogram  

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