Halving a square

Now the opposite problem. Given a square, make one with half the area.

The temptation is the same as before, and wrong in the same way. A square with half the side has area ½ × ½ = ¼ of the original: four of them would fill it, not two.

The right move is simply to run the doubling construction backwards. If building outward on a diagonal doubles, then drawing inward should halve.

Draw a tilted square inside the larger one, with its corners at the midpoints of the larger square's sides.

Why is that one half? Add the east-west and north-south lines again. The four corners of the big square that lie outside the tilted one are four small triangles; the tilted square itself is made of four more of the same triangles. Four against four, so the inner square is exactly half.

With paper it's quicker than describing. Cut out a square and fold each corner inward so that the creases run through the midpoints of the sides (the corners meet at the centre). The square PQRS you're left with is the one you want.

To prove it properly, join QS and PR, work out the angles those lines make, and use triangle congruence to show all eight small triangles are the same.

Notice that this is the doubling construction seen from the other side: the big square is built on the diagonal of the small tilted one. Same picture, read in the opposite direction.