I have the next quadrilateral with midpoints E F G H. The source of the problem is my class of geometry, I read the book but I don't find anything related to this.
I found in Google about Varignon's Theorem then I do the following construction:
My question is: it is correct to state that quadrilateral ABCD is a parallelogram? I think that is wrong because I don't use the areas of the statement.
Let ABCD be a rhombus and EFGH be the rhombus formed by mid-points of sides. Any projection of ABCD on an arbitrary plane has its area reduced by a factor $\cos \theta$ where this angle is between their plane normals.
The inner rhombus area is one-fourth the outer among all the possibilities.