An Astonishing Property of 3rd Order Magic Square

In an article written in the February 1999 issue of the American Mathematical Monthly, titled appropriately “Magic Squares Indeed!”, the authors point out a truly remarkable property of this magic square; namely:
8162 + 3572 + 4922 = 6182 + 7532 + 2942,
8342 + 1592 + 6722 = 4382 + 9512 + 2762.
Please verify for yourself that these relations are true. In fact, there are even more such relations, but we will list them later. For the moment we ask: what makes these relations true?
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