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,
AC² + BD²
= AB² + CD² + DE² + AF² - AE² - DF²
= AB² + CD² + (AD - AE)² + (AD + DF)² - AE² - DF²
= AB² + CD² + 2 AD² + AE² - 2 AD AE + DF² + 2 AD DF - AE² - DF²
= AB² + CD² + AD² + CB² + 2 AD (DF - AE) as CB = AD..
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...