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This problem trains for: AMC-10, SAT-II, AMC-12.

In a parallelogram, express the sum of the squares of the lengths of the two diagonals as a function of the lengths of the sides. All the lengths are labeled as in the figure:

figure

Apply the law of cosines in the triangles ABD and ABC. Notice that the opposite sides of the parallelogram are equal in length. Also, the adjacent angles are supplementary:

figure

equation

equation

Use the identity:

equation

and now add the two equations:

equation