Splattered paint

A partial solution:

Consider a vertical line swiped from the left to the right. Slowly the amount of yellow on the left side will increase from zero to the total yellow area, and the amount of green on the right size will decrease from the total green area to zero. At some point they will meet, and the amount of yellow on the left side will be the same as the amount of green on the right side. Note : The orientation (vertical) we chose for the line was arbitrary, so WLOG any angle could have been chosen.

The same argument holds for magenta and blue.

In general, the place where the green/yellow and magenta/blue lines meets will be different for a different orientation.

However, as the orientation is rotated from $0^\circ$ to $360^\circ$ , at some point, the lines will meet in the same place for the green/yellow combo as it does for the magenta/blue combo. Now, to come up with a more rigorous proof as to why this is so... Stay tuned!

First, consider a line cutting across the canvas at a given angle. Then, as how Geoff describes this, moving this line from side to side will decrease the amount of one color while increasing the amount of the other, both varying monotonically. There will be a point of intersection for both pairs of colors, but not necessarily at the same place, i.e., two different lines will be necessary, not one, usually.

We make up a configuration plot , as shown in the graphic above. The outer black circle represents one extreme of line displacement while the inner black circle represents the other extreme. The circular plot represents the different angles the line cuts across the canvas. The gray circle represents the exact halfway between the two extremes. The point where the monotonically varying functions of pairs of colors intersect are represented by two different curves, shown here in blue and green, for magneta-blue and yellow-green pairs, respectively.

Let us say that at a given angle of the line cutting across the canvas, where the monotonically varying functions of pairs of colors intersect, a certain percentage of one color is found on the left side of the line. This means that when this angle is changed by 180 degrees, then it's the complement percentage of that color is now found on the left side of the line. In other words, both the blue and green curves have the property that for every point on it, there is an antipodal point that is at an equal-but-opposite distance from the center gray circle. This will force an intersection between the blue and green curves, and that's where the line (gray) will be found that cuts both pairs of colors in the manner described.