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See diagram...

we apply the principle of Pythagoras..

BE² + DE² = BD² --- (1) AF² + CF² = AC² --- (2)

BE² = AB² - AE² CF² = CD² - DF² so substituting them in the equations above,

In triangles ABE and CDF, the angles BAE and CDF are equal.. AB = CD. BE and CF are equal , as they are distance between parallel lines. Hence, AE = DF as they are two congruent triangles...

Thus, DF = AE..

So, sum of squares of diagonals = sum of squares of sides...