1. An automatic cell analyzer comprising: an imaging unit which captures fluorescence emitted from a cell and which acquires a cell image; an exposure changing section which changes an exposure condition when the imaging unit captures cell images; and a processing section which analyzes the cell based on a plurality of cell images respectively captured under the changed exposure conditions; wherein said exposure changing section changes between: (i) a first exposure condition for capturing a first cell image with which a shape of a thin projecting part of the cell can be analyzed, and (ii) a second exposure condition for capturing a second cell image with which a shape of another part of the cell other than the thin projecting part can be analyzed; and wherein said processing section is configured to detect said thin projecting part based on a diameter of an inscribed circle inscribing a profile shape of each of said cell images.