1. A projector comprising: an image pickup unit which obtains an image of a subject having a projection display area; a distance measuring unit which measures a distance to the subject: an image signal processing unit which: cuts a projection display image corresponding to the projection display area of the subject contained in the image of the subject obtained by the image pickup unit to produce a cut image signal, produces a size image signal showing the size of at least a portion of the subject based on the measured distance to the subject, combines the cut image signal and the size image signal into a combined image signal such that the size image signal is superimposed on the cut image signal in the combined image signal, and processes the combined image signal such that the combined image has a predetermined size to produce a projection image signal; and a projection unit which produces an image according to the projection image signal produced by the image signal processing unit and projects the image.