# Orthogonal 4-cut of a convex polygon

Given a convex polygon with N vertices I need to cut it into four equal area parts with two straight orthogonal cuts.

I feel that I have all the necessary pieces to solve this puzzle, but I can't put them together. For example, I could cut this polygon in two equal area parts, yielding two polygons which I know can be cut in halves by a single cut according to the ham-sandwich theorem. But how would I make the first cut? It seems to me that there is a multitude of ways to cut a polygon in halves, so which one should I choose?

I could find the centre of area of this polygon - but would it be the point at which these two cuts intersect? If it is, again there may be a multitude of cuts through that point which divide the original polygon in halves. Should I just try all of them and see if the orthogonal cut does what I need? Or is there a simpler solution (assuming, of course, that the cuts I'm looking for intersect at the centre of area, which is not at all obvious to me)?

• I'm trying to solve the same problem. Are you working with a polygon on an integer grid, or an arbitrary one? – Chiffa Aug 14 '14 at 10:18