When is proving a lemma necessary?

Question

The following diagram is often used to illustrate one of the many proofs of the Pythagorean Theorem.

http://www.mathsisfun.com/geometry/images/pythagorean-theorem-proof.png

But first, I think a lemma needs to be proven showing that the construction is even possible, and specifically that the quadrilateral with sides of length "c" is indeed a square. It needs to be a square to prove the Pythagorean Theorem.

No website that explains the proof ever mentions the lemma. When is proving a lemma, first, necessary?

Answer 1

The picture enough is proof that the construction can be done (unless you mean construction as in geometric construction with a straightedge and a compass, but that isn't necessary here).

Also, it's fairly evident from facts about geometry that the inner quadrilateral is a square. If you start measuring all the angles, you'll see why.

For example, on the bottom line segment of the larger quadrilateral, there are three angles (vertex at A) - a left one, a middle one, and a rightmost one. The left one and rightmost one add up to 90 degrees, because the rightmost angle is equal to the bottom angle on the left side of the quadrilateral (with vertex D), and the sum of the angles in a triangle is 180 degrees. Thus the middle angle with vertex A must be 90 degrees. This holds true for all sides of the outer quadrilateral, so the inner quadrilateral must be a square.

This picture is enough to prove the Pythagorean Theorem, but of course it wouldn't hurt to throw in a few explanatory words here and there. However, I think proving things as lemmas is a bit overkill, because the results you need to 'prove' are immediate applications of theorems in planar geometry (so it would be a one-step proof really..)

Hope this helps.

Pascal: True, in general, you need to be careful about drawing conclusions from diagrams. But what I was saying is that in this case, the diagram is enough (with a few basic results from geometry that can be readily applied by studying the diagram).

Answer 2

You are correct. You DO need to show that the resulting figure is a square. The proof of this relies on the fact that the corner of the square plus the two acute angles of the triangle together equal a straight angle (the side of the large square), and that the two acute angles of the triangle together sum to 90°. This is in turn derived from the fact that the sum of all the angles in a triangle is 180°, which is equivalent to the parallel postulate. If the parallel postulate fails, you will be unable to perform this construction. Indeed, you will not be able to create a square of any dimension -- the existence of a single rectangle is equivalent to the parallel postulate. So is the Pythagorean theorem itself for that matter.

Re. Chris W: actually, you should be careful about assuming a construction can be done simply because the diagram says so. Check out the "proof" that all angles are zero at http://en.wikipedia.org/wiki/Invalid_proof#Examples_in_geometry

Answer 3

In response to the question about a lemma. In general, lemmas are intermediate results that are used to prove another result. It is necessary to prove the lemma if you use the result of the lemma to argue a result of another Theorem.

For example, when proving the sqrt(2) is irrational (or more correctly, disproving it is rational), one uses the intermediate result (or lemma) that if the square of a number is even, then the number is even.

So, from this example, one can see that what is called a lemma is a function of the ultimate result one is trying to prove.