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MIT Mathematician confirms: Israeli 10th-grader discovers new geometric theorem

israelhayom.com

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Re: MIT Mathematician confirms: Israeli 10th-grader discovers new geometric theorem

#21

Post by jschulenklopper »

hydroo wrote: post #7
jschulenklopper wrote: post #3

From the article: > According to the new "Three Radii Theorem," if three or more lines extend from a single point to the edge of a circle, then the point is the center of the circle and the straight lines are the radii. Shouldn't that be "three or more lines of equal length ", and "to different points on the edge of a circle"? Or am I missing something?

The theorem as stated in the article is definitely wrong. Equal length would make it correct, I think.

> Equal length would make it correct, I think.

And the three lines of equal length should go to different points on the edge of the circle as well. Three lines from a point to the same position on the edge wouldn't work - but perhaps I'm mathematically paranoid now :-)

Re: MIT Mathematician confirms: Israeli 10th-grader discovers new geometric theorem

#23

Post by bo1024 »

jschulenklopper wrote: post #3

From the article: > According to the new "Three Radii Theorem," if three or more lines extend from a single point to the edge of a circle, then the point is the center of the circle and the straight lines are the radii. Shouldn't that be "three or more lines of equal length ", and "to different points on the edge of a circle"? Or am I missing something?

That was my thought too. Here's my proof: Let P be the point, r the length of the lines, and A,B,C the other endpoints of the lines.

A, B, and C all lie on the circle of radius r centered at P. The question is if they can simultaneously lie on some other circle. In other words, the question can be restated as whether two distinct circles can intersect at three or more points.

One way to see that this is impossible is to consider the equation of a circle. There are three parameters (for instance, x and y coordinate of the center along with the radius). Hence, by specifying three points on the circle, one creates a system of three equations with three unknowns, which has a unique solution.

For the generalization to dimension d (where in the original example d = 2), then I think this shows that d+1 equal-length lines from a point to a hypersphere imply that the point is the center.

Re: MIT Mathematician confirms: Israeli 10th-grader discovers new geometric theorem

#24

Post by keehun »

nthcolumn wrote: post #8
jschulenklopper wrote: post #3

From the article: > According to the new "Three Radii Theorem," if three or more lines extend from a single point to the edge of a circle, then the point is the center of the circle and the straight lines are the radii. Shouldn't that be "three or more lines of equal length ", and "to different points on the edge of a circle"? Or am I missing something?

And wouldn't two of equal length do? I think we are missing an explanation of the theorem. edit: oh hang on, I see now: Trump Foundation It's one of those theorems. edit: yes three mean they can't be longer than the radius. Isn't it interesting. You need 3 for a circle. What about other shapes?

This is what could happen with only two lines. An approximate drawing, but you get the idea.[0]

[0]: http://imgur.com/gDjm3AS

Re: MIT Mathematician confirms: Israeli 10th-grader discovers new geometric theorem

#26

Post by mankash666 »

Seems like the kind of problem solved routinely in city level Olympiads world wide. I believe the kid is certainly bright, but by this article's standard, most Olympiad problem sets, even at the city level, become new theorems. So, keep the enthusiasm going kid, but try your luck at the Olympiads to truly benchmark your standing.

Re: MIT Mathematician confirms: Israeli 10th-grader discovers new geometric theorem

#27

Post by hasenj »

jschulenklopper wrote: post #21
hydroo wrote: post #7

Earlier quoted context omitted.

The theorem as stated in the article is definitely wrong. Equal length would make it correct, I think.

> Equal length would make it correct, I think. And the three lines of equal length should go to different points on the edge of the circle as well. Three lines from a point to the same position on the edge wouldn't work - but perhaps I'm mathematically paranoid now :-)

Well they wouldn't really be three lines if they all go from the same point to the same other point. They would all be the same line.

Unless the language of math does allow "references" like programming languages.

All you need is to require three different lines of equal length.

But I don't see the significance of calling this a theorem. It seems perfectly obvious and elementary and almost just the definition of a circle.

Re: MIT Mathematician confirms: Israeli 10th-grader discovers new geometric theorem

#28

Post by javajosh »

catpolice wrote: post #17
eximius wrote: post #2

Hm, a clever little theorem. I'm surprised it isn't recorded somewhere. Perhaps it's just under a different name because of it's relative... simplicity? Not to belittle her accomplishment. This is something you'd expect Euclid to write about or something.

It follows from the statement that if two circles intersect in more than two points, they're identical. Which seems like a familiar theorem, that I can't seem to place. So it's more like a corollary. It would be remarkable if this hadn't come up before, but I want to believe because of how good a story it makes...

Ha, that's a good one, actually. Quite elegant.

Re: MIT Mathematician confirms: Israeli 10th-grader discovers new geometric theorem

#29

Post by hasenj »

Until someone brings a link from MIT's website, I'm calling bullshit on this one.

This sounds like what pops up every other day in Egyptian newspapers about genius Egyptian kids who invent this or that.

The theorem stated in the article is not a theorem at all. It's a direct consequence of the definition of a circle and is perfectly obvious to anyone who spends two minutes pondering the implications of that definition.

Re: MIT Mathematician confirms: Israeli 10th-grader discovers new geometric theorem

#30

Post by noobie »

adt2bt wrote: post #25
randlet wrote: post #19

Kudos to the teacher who helped her develop her idea instead of scolding her for using ideas that weren't being taught in his classroom.

How common is that?

scolding [a student] for using ideas that weren't being taught in his classroom.?

Very common!

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