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Georgia Teen Stumbles On New Theorem
dread minerva writes "Proof that the kids are alright: The Atlanta Journal-Constitution published the following article about Josh Klehr, who discovered a math theorem while sitting in study hall one day in eigth grade. The theorem is now known as the Klehr-Bliss Theorem and a paper on it is being published in The American Mathematical Monthly."
He was not suspended, expelled, or arrested for "Thinking While In Highschool"!
I AM, therefore I THINK!
First and foremost, I don't think the theorem is actually called the Klehr-Bliss theorem. AFAIK it's the van Lamoen theorem, since he was the first to furnish a full proof. Lou Talman had a quicker (and simpler) proof that was purely geometric, but I believe it was found to be flawed. I was working on a brute-force algebraic manipulation, but Floor van Lamoen carried essentially the same technique to its completion before I was able to. You can read about his proof here.
Josh's conjecture was pretty accurately summarized in the article. The point E mentioned is actually the circumcenter, the center of the only circle passing through the three vertices of the triangle. Also, it is not exactly correct to say that the lines through A, B, and C intersect in "a point" inside the triangle. The three lines are concurrent (they all pass through a common point, a rare thing for three lines to do), but Josh's slope-reciprocal construction is really just a reflection about the line y=x in the coordinate plane, and changing the orientation of the coordinate axes relative to the triangle makes the point of concurrence wander around inside the triangle. The kicker that I noticed is that as it wanders, it stays on the nine-point circle, or Feuerbach circle of the triangle. I've actually found that there's a lot more to be said along these lines, and to my knowledge none of it has been published.
For the public/private thread... I think that Josh was and is attending a private school (Paideia, an excellent school by the way) though I attended a public one (Collins Hill... not too bad as public schools go).
Not only does the theorem have little to no practical value, it also is of little interest to mathematicians. I've always thought of it as simply a little ditty in triangle geometry. I haven't yet read van Lamoen's article in the AMM, but I believe he mentions it only in passing.
And yes, it is vitally important to have an encouraging mentor. Steve Sigur, Josh's teacher, is a great guy and an excellent math teacher. I don't mean this to trivialize Josh's accomplishment--it's also vitally important to have a creative mind and be willing to explore--but Mr. Sigur deserves the real praise here.
I'd also like to take this opportunity to shamelessly plug The Geometer's Sketchpad. It's a great piece of software that dynamically creates geometric constructions. It's excellent for visualization. I used it to see the generalizations I was after, and I think Josh was using it when he first made his conjecture. If you've any interest in geometry--or are willing to have some anew--you should check this out. You can download a free sample version.
Anyway, I just wanted to post and settle a few things... If anyone has any questions, you can post them here or email me (I'm abliss at freeshell.org). Thanks for your attenton!
http://home.wxs.nl/~lamoen/wiskunde/concur.html
The extremely vague statements in the article look similar to what is presented there...
I hope he gets an A in math for the rest of high school. How I would love to be in his math class and hear him say to his teacher "when you come up with law of mathematics on your own then you can tell me my math homework is wrong!"
I've hit Karma 50 and gotten a Score:5, Troll... I win!
The Monthly article in question appears to be "Morley Related Triangles on the Nine-Point Circle", by Floor van Lamoen, Amer. Math. Monthly vol. 107, no. 10, Dec. 2000, pages 941-945. The introduction says: "We identify two points M and H on Euler's nine point circle CN, found as intersections of three reflected lines. M and H each depend on the direction of a set of parallel lines. Posing the condition that M and H coincide for a certain direction, or that MH is a diameter of CN, we find two equilateral triangles in CN homothetic to Morley's famous trisector triangles."