Note that $GA + GB > AB$ and $GH + GL > HL$, so $AH + BL = (GA + GH) + (GB + GL) > AB + HL$. Analogously, $BI + CG > BC + IG$, $CJ + DH > CD + HJ$, $DK + EI > DE + KI$, $EL + FJ > EF + LJ$, and $FG + AK > FA + GK$. If we assume $FAHJ$ is circumscribed around a circle, then we have $BI + AK = AB + KI$, $CJ + BL = BC + LJ$, $DK + CG = CD + GK$, $EL + DH = DE + HL$, $FG + EI = EF + IG$, and $AH + FJ = FA + HJ$. Adding these inequalities and adding these equations, we get a contradiction. So the answer is no. $\Box$