We discuss a proof-of-principle experiment in which we detect triple electron emission from a surface due to primary electron impact. The new aspect is the ability to record the energies and emission directions of the ejected electrons. We selected NiO films as a target, which have shown in previous electron pair emission studies to give an enhanced intensity compared to other materials. The triple sum energy spectrum displays a shape consistent with a self-convolution of the electronic density of states. We define two different emission geometries. While the energy distributions are essentially identical, the intensity levels differ by a factor of 2. Imposing a geometrical constraint on one of the emitted triples shows that the available energy is equally shared among the other two electrons. We discuss our findings within a simplified scattering model. We also present angular distributions. Prominent intensity minima for electron emission in the same direction are not observed in contrast to our previous electron pair emission studies.