Explore the quadrisections of a polygon.
P is a convex polygon whose vertices are the (nonzero except possibly the first) columns of the 2 by 6 array below. The graphic (which may take a minute or so to generate) shows the polygon together with any Bernoulli Axes (A magenta diameter paired with a perpendicular diameter which cuts the trapezoid into 4 equal areas. The oval is the Bernoulli Oval, the centers of the perpendicular pairs of diameters. It often but not always circles the centroid (the black point). The red point is the average of the vertices. Experiment and make a conjecture.