Can a circular disc be 'dissected' by a rectangular grid into a finite number of pieces in such a way that the individual pieces of the circle can be grouped into regions of equal area?
Clearly this can be done in some cases:
(1) One horizontal and one vertical line crossing at the circle center
(2) Only vertical (or horizontal) lines dividing the circle into regions of equal area
(3) Case two with one line through the circle center perpendicular to the other lines
Can this be done, however, with at least two horizontal and two vertical lines cutting the circle?
For example, can it be done with exactly two horizontal and two vertical lines tic-tac-toe style? (I suspect not but do not have a proof.)
The following "proof without words" might be useful:
Yes, it can be done in some cases. Your example of two vertical and two horizontal lines is one. Following the tic-tac-toe board, there are four corner pieces (like your B), four edge pieces (like your A), and one center piece. If we place the lines symmetrically around the center and make the area of the center piece $\frac 15$ of the disk, the center will match the sum of one edge and one corner like $A' + B'$. This cuts the disk into five pieces of equal area. For a disk of radius $1$, the spacing would be $\sqrt{\frac \pi 5}.$ Another possibility is to have three horizontal and three lines. You create four center pieces, eight edges, and four corners. If we make each center piece $\frac 18$ of the disk, the centers will match the sum of two edges and one corner, giving eight pieces of equal area. The center lines go through the center of the disk and the side ones are spaced $\sqrt{\frac \pi 8}$ away.