BALTIC-WAY 1992 Problem 5

Given that $a^{2}+b^{2}+(a+b)^{2}=c^{2}+d^{2}+(c+d)^{2}$, prove that $a^{4}+b^{4}+(a+b)^{4}=c^{4}+d^{4}+(c+d)^{4}$.