notice $2(a^4+b^4+(a+b)^4)=(a^2+b^2+(a+b)^2)^2$ equality follows

Oh, I remember proving the above identity when solving a descartes' circle theorem question. It's a pretty cool identity.

jasperE3 wrote: Prove that if real numbers $a,b,c,d$ satisfy $a^2+b^2+(a+b)^2=c^2+d^2+(c+d)^2$, then they also satisfy $a^4+b^4+(a+b)^4=c^4+d^4+(c+d)^4$. https://artofproblemsolving.com/community/c4h1939863p13375242