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Quadrilateral fill in the blank (Posted on 2018-01-25) Difficulty: 3 of 5
A quadrilateral is a _________ if and only if the sum of the squares of its sides is equal to the sum of the squares of its diagonals. Prove it.

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A draft for solution Comment 2 of 2 |
If a quadrilateral verifies that property then the middle points of both diagonals are a single point (applaying Euler's theorem for quadrilaterals), and this implies that it is a paralelogram. 

D^2+d^2 = a^2+b^2+c^2+d^2+4MN 
where MN is the segment uniting the middle points of both diagonals. 
Also for the theorem of cosine it is very easy to show that each paralalogram verifies that property. 

  Posted by armando on 2018-01-27 09:50:42
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