Chemically active droplets are nonequilibrium systems that combine phase separation with chemical reactions. We here investigate how the activity introduced by the chemical reactions influences solid particles inside such droplets. We find that passive particles are centered in active droplets governed by first-order reactions. In autocatalytic active droplets, only catalytically active particles can be centered. An example of such systems in biology are centrosomes. Our study can account for the observed positioning of centrioles and provides a general mechanism to control the position of particles within chemically active droplets.