The authors investigate the second order delay differential equation \((d^ 2x/dt^ 2)(t)=f(x(t-\tau))-ax(t),\) where f is a decreasing step function with only one step. Normalization reduces the problem to the standard form, \((dx/dt)(t)=y(t)\), \((dy/dt)(t)=f(x(t-\tau))-x(t)\), \(f(\xi)=1/2\) if \(\xi\leq \theta\), \(=-1/2\) if \(\xi >\theta\), with constants \(\tau >0\) and \(\theta\in {\mathbb{R}}\), and with initial values \((x_ 0,y_ 0)\) given by some \(y_ 0\in {\mathbb{R}}\) and a differentiable function \(x_ 0: [-\tau,0]\to {\mathbb{R}}\) satisfying \(x_ 0(t)=\theta\) for at most finitely many \(t\equiv [-\tau,0]\). It is proved that, in principle, the solutions can be constructed geometrically by compass and ruler. (This construction reminds of similar techniques in solving certain two- dimensional linear time-optimal feedback control problems.) In this way, analytical results on existence of periodic orbits and limit cycles are obtained. Moreover, computer simulations for varying parameters \(\tau\) and \(\theta\) reveal a remarkable richness of different types of periodic orbits, and lead to conjectures on their stability and bifurcations.