Investigation: Squares on the sides of Parallelograms, Etc.

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• On a parallelogram P construct exterior squares on each side. Prove the centers of these squares form a square.
  
  

Try some auxillary lines. Suggestion?

• On a parallelogram P construct squares over the interior on each side. Prove the centers of these squares form a square.
  
  

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  Suggested Auxilliary Figure?

Extension

• Do both of the constructions above on the same parallelogram and call the area of the square formed by the centers of the externally constructed squares on the sides to be E, the area of the square formed by the centers of the squares constructed to the interior to be I, and the area of the parallelogram to be P. Use GSP to form a conjecture about these three areas and then prove your conjecture. Hint.

• Generalize the relationship among the areas of the resulting quadrilaterals when P is an arbitrary quadrilateral. Click here for a GSP sketch (you may be better off building your own).
  
  
• Explore the situation when equilateral triangles are constructed on each side of the parallelogram P and a resulting quadrilateral is formed by the centers of the triangles.
  
  
• Explore the situation when equilateral triangles are constructed on each side of the parallelogram P and a resulting quadrilateral is formed by the vertices of the triangles.