https://aops.com/community/p11377651 rmtf1111 wrote: First of all, let's replace the ugly 3's with ones(i.e, let $ab+bc+ac=de+df+ef=\cdots=1$). Let $a=\tan{x_1} , b=\tan{x_2} ,\cdots ,i=\tan{x_9}$. Note that $x_1+x_2+x_3=\frac{\pi}{2} \ , \ x_4+x_5+x_6=\frac{\pi}{2} \ , \ x_7+x_8+x_9=\frac{\pi}{2} \ , \ x_1+x_4+x_7=\frac{\pi}{2}$ and $x_2+x_5+x_8=\frac{\pi}{2}$, thus $x_3+x_6+x_9=\frac{\pi}{2}$ also, hence $cf+fi+ic=1$.