Riddler Classic: rectangles

This week's Riddler Classic is slight variation on a classic puzzle. In fact, it's a special case of the classic puzzle, but despite that, I think is probably actually harder than the original. Inspired by James Barton's 15 minutes of fame last week, I thought I'd write up my solution to this one. 

Here's the puzzle: 

Say you have an “L” shape formed by two rectangles touching each other. These two rectangles could have any dimensions and they don’t have to be equal to each other in any way. (A few examples are shown below.)
Using only a straightedge and a pencil (no rulers, protractors or compasses), how can you draw a single straight line that cuts the L into two halves of exactly equal area, no matter what the dimensions of the L are? You can draw as many lines as you want to get to the solution, but the bisector itself can only be one single straight line.

How to go about this? Well, the first thing to notice that any line which goes through the centre of an individual rectangle bisects the area of that rectangle exactly. To see this, consider what happens if you rotate the rectangle through 180 degrees - the line will meet itself, and the two pieces will sit on top of one another. So not only does every line through the centre of a rectangle divide into two pieces of equal area, it actually divides it into two congruent pieces. 

So how does that help us? The key is to realise that an 'L shape' is actually two rectangles - one filled in, and one blank. It's easy to construct the sides of the 'blank' rectangle by just extending top and right hand side of the 'L' (the dotted black lines in the right hand diagram). You can then find the centres of both rectangles by drawing their diagonals (the blue and red crosses). 

Now, the line which passes through the centres of both of these rectangles bisects the area of the big rectangle and also bisects the area of the small rectangle, so it must bisect the area of the L shape as well (half of the big rectangle is equal to half of the small rectangle plus half of the L). 

Ok, nice puzzle - but what about the the classic puzzle I mentioned earlier? Well, there's no reason in this puzzle for the little rectangle and the big rectangle to be lined up so neatly. (in fact, there's not really any reason for them to be anywhere near each other). The 'classic' version of this puzzle that I've heard before is about cutting a cake with special frosting on one section, and dividing the special frosting exactly in two, for which the solution is pretty much exactly the same. 

There are some interesting extensions to this new puzzle. It's pretty clear that wherever you put the icing on an rectangular cake, and whatever shape it is, you can always divide both cake and icing exactly in two with a single straight cut (rotate the knife around the centre of the cake, it will eventually pass over the whole of the icing, so at some point it must divide it exactly in two), although you won't necessarily be able to construct this with just a straightedge. But for what other shapes of cake and icing is it possible to do this in general? And when you do need the icing to be carefully placed to allow it to work? 

As I said, I think the L shape puzzle is actually probably harder than the more general version, because the way the rectangles are arranged makes it possible to draw so many more lines - if you're given the 'tilted' version, there's basically only one thing you can try with a pencil and straightedge. With the 'L shaped' version, it's possible to go down a variety of dead ends joining up corners of the L with each other before you get to the answer.