Consider the labeling for which \(\displaystyle \sum_{i=1}^n P_iQ_i\) is the maximal. Then it has the desired property. Indeed, for the sake of contradiction suppose that there are indices \(i\ne j\) such that the balls with diameters \(P_iQ_i\) and \(P_jQ_j\) and centers \(O_i\) and \(O_j\) do not have a common point. Then \[\frac{P_iQ_j+P_jQ_i}{2}\ge\left|\frac{\overrightarrow{P_iQ_j}+\overrightarrow{P_jQ_i}}{2}\right|=|\overrightarrow{O_iO_j}|>\frac{P_iQ_i+P_jQ_j}{2}.\]This contradiction finishes the proof.