THEOREM B: Every simple closed curve has lots of inscribed parallelograms and lots of inscribed rhombuses. In fact, if J is any simple closed curve and L is any line then J has an inscribed rhombus with two sides parallel to L.

Proof: First of all, we may as well assume that the line L is the x-axis, for we can choose our coordinate system any way we want to. We need to show that J has an inscribed rhombus with two horizontal sides.

The proof uses a technique we might call "mountain climbing". Imagine the curve J as representing two faces of a "mountain". The base of the mountain will be at some point P with minimum y-coordinate, while its summit will be at a point Q with maximum y-coordinate. (For simplicity, we will assume that the minimum y-coordinate is zero so that the point P is actually on the x-axis L.)

Imagine four mountaineers on this mountain. Two will start at the base P and will climb upward (one on each face of the mountain) so that their elevations (y-coordinates) always remain equal. A formal proof that this is possible is rather messy, but it should be intuitively believable. Sometimes, of course, one of the pair will have to back up a bit to allow the other one to decend a hill before they can both begin ascending again.

The other pair of mountaineers will begin at the summit Q and will repel down the mountain (again, one on each face) so that their elevations always remain equal.

The next step will require even greater coordination between the four mountaineers! Let's have the two pairs coordinate their efforts so that the distance between the ascending pair is always equal to the distance betwen the descending pair. This might require one of the pairs to occasionally reverse their course momentarily, but again it can be proved that this is possible.

As a result of this coordination, the four mountaineers' locations always form a parallelogram inscribed in J, and two sides of the parallelogram are horizontal. Initially the parallelogram is very tall and skinny, but eventually it will become very short and fat (near the moment when the pairs pass each other on their routes up and down the mountain). At some time in between, the parallelogram must be a rhombus!