Summary: We use a planar analysis approach to show the existence of periodic solutions of the system of delay differential equations \(x'(t)=dL [x(t-\tau)-x(t)] +f(x(t))\), for \(d\), \(\tau>0\), \(L\) a positive semidefinite \(2\times 2\)-matrix, and \(f\) is a smooth nonlinear function satisfying some negative feedback conditions. This involves finding a cone \({\mathcal K}\) in the phase space \(C([-\tau,0],\mathbb{R}^2)\) such that every solution \(x_t(d,\varphi)\) of the equations subject to \(x(t)|_{[-\tau,0]}= \varphi (t)\) with \(\varphi\in{\mathcal K}\), returns to \({\mathcal K}\) after some time \(0