A detailed analysis of multicolour Geneva photometry and high-resolution echelle spectroscopy of the B8.5V star V945 Cen (HD112409, V = 5(m).2) whose short-term periodic variability was first detected from the HIPPARCOS data, led to the finding that the object is an ellipsoidal variable and a close double-lined spectroscopic binary with a 0(d).6496 orbital period and a circular orbit. The components have spectral types B7 and B8.5 and are both main-sequence objects. The system is detached but close to a contact configuration and is observed at an inclination of similar to 23-24 degrees. A notable finding is that the optical spectrum also contains a third system of spectral lines corresponding to a late B spectral type that remained stationary during the 5 days of spectral observations. From several independent lines of reasoning, we tentatively conclude that the third spectrum belongs to a real third body in the system. However, the possibility that it instead originates in circumbinary gas cannot be excluded completely, and additional spectral and interferometric observations are required to help us determine its origin conclusively.