Given a 2D convex polygon P and its centroid C, how do I find the longest line segment passing through C, where the endpoints of the segment lie on the boundary of P?

Intuitively I imagine there are only a small number of possibilities, so I could just compute the lengths of each of those line segments and see which is longest. For instance, I might check each such line segment passing through C where at least one endpoint is a vertex of P.

Background: I would like simulate an epithelial tissue represented as a collection of Voronoi cells. Per http://dash.harvard.edu/bitstream/handle/1/4731601/Gibson_Control_Mitotic.pdf the mitotic cleavage plane during cell division lies along the "long axis", which I define above as the line segment I would like to find.