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It's called a "midpoint polygon". The problem seems to be solved addressed (with proofs for triangles and quadrilaterals but only a conjecture for the pentagon) here:

http://techhouse.brown.edu/~mdp/midpoint/pentagons.php

(The other construction mentioned is a {5/2} star polygon inscribed in a pentagon.)

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It's called a "midpoint polygon". The problem seems to be solved here:

http://techhouse.brown.edu/~mdp/midpoint/pentagons.php

(The other construction mentioned is a {5/2} star polygon inscribed in a pentagon.)