I don't think it is obvious how to find the shortest chord that bisects the area of a convex polygon. That is the question posed in this now closed post: Shortest bisecting line. Perhaps one would have to use the algorithm below, modified to spin the direction through $180^\circ$.
Shermer, Thomas C. "A linear algorithm for bisecting a polygon." Information processing letters 41, no. 3 (1992): 135-140.