Proving a Convex Quadrilateral is a Square with Internal Point O as its Center

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To prove that a convex quadrilateral PQRS is a square with an internal point O as its center, it is necessary to demonstrate that the equation 2A = OP^2 + OQ^2 + OR^2 + OS^2 holds true. The discussion emphasizes the importance of visualizing the problem through a diagram and applying the Pythagorean Theorem to understand the relationships between the sides of the quadrilateral. It suggests that bisecting segments and squaring the results can provide insights into the relationship between the segments and the area of the square. The connection between the lengths OP, OQ, OR, and OS is crucial in establishing the proof. Ultimately, proving this relationship confirms that PQRS is indeed a square with O at its center.
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Homework Statement



Let O be an internal point of a convex quaddrilateral PQRS whose area is A.

Prove that, if 2A = OP^2 + OQ^2 + OR^2 + OS^2, then PQRS is a square with O as its centre

Homework Equations





The Attempt at a Solution



I have no idea where to start, except that I know that I need to use 1/2 ab sin C
 
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Forget the 1/2 ab sin C for now. Draw a picture and think Pythagorean Theorum.
 
how does PQ, QR, RS and SP relate? Are they the same or different? If you bisect the segment PQ and square the result, how does that relate to the square of the segment OP? How about to the area of the square?
 

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