pythagorean theorem proof

Factual Questions
21

imonacomputer,

One of the ways an author captures my attention is by providing several proofs of the same theorem. Thanks to all those that do. Unfortunately, I think publishers eschew this as unnecessary or wasteful; one of the problems of balancing business with scholarship, I suppose. However, published by the Mathematical Association of America, there are the wonderful books Proofs Without Words and Proofs Without Words II, both by Roger Nelson. Each contain … just checking … six proofs (without words) of the Pythagorean theorem. Both are currently available and worth the time to look through, especially if you like geometry.

As for the ‘8 year-old’ question, I’m at a loss. IMHO, I’m inclined to believe zut is on the right track. If the Pythagorean theorem (PT) was proven recently, it was probably in response to the statement of the theorem. There’s a world of difference between proving the PT and discovering the PT, and the former, I’m fairly certain, would be within the capabilities of many prodigious eight-year-olds.

Sincerely,
Pies ‘R’ Squared

22

When I first read it, I had no idea what the OP was referring to. Today, however, I ran across the following, which I believe is quite possibly what the OP was referring to (it was on the front page of my local paper this morning, a Cox News Service article, so I’m surprised no one else has mentioned this today).

According to the article, in May of 1999 an Atlanta high school student, Josh Klehr, discovered a new geometrical result. He showed it to his teacher, Steve Sigur, and it was recently published in the American Mathematical Monthly and given the name the Klehr-Bliss Theorem (Bliss is another Atlanta area student who did some follow up work).

The description of the result in the paper is rather poor, but I believe I can figure out what it’s trying to say (I haven’t checked this work myself, so I can’t say positively I interpret it correctly).

OK, so a triangle has various points, or “centers”, associated with it. For example, all the altitudes of a triangle intersect in a certain point; all the bisectors of the angles of a triangle intersect in a certain point; the perpendicular bisectors of the sides intersect in a certain point; and so forth.

Klehr discovered a new such point. For background: Given a line with slope m, any line perpendicular to the given line has slope -1/m (negative reciprocal of the original slope). Klehr played around with lines with reciprocal slope (not negative). I’ll call these “reciprocal” lines. So, for a simple example, a reciprocal line to a line with slope 3/7 would have slope 7/3, of course.

So start with a triangle, and construct reciprocal lines to each of the three sides of the triangle (it’s not clear from the article where the lines should be constructed, but I tend to assume they mean “reciprocal bisectors” of the sides). These reciprocal lines will intersect in a single point, the new point Klehr found.

23

do you or anyone else know a web-site that has more than just the 34 proofs of the pythagorean theorem, because if u do it would help a lot, and i found 1 copy of the book that everyone is saying has 300+ proofs and it was made in 1927, but i haven’t gotten to look at it yet so i hope it’s the right one, thanx