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MIT Mathematician confirms: Israeli 10th-grader discovers new geometric theorem
- eximius 11y agoHm, 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.
- mchahn 11y ago> This is something you'd expect Euclid to write about Which is what makes the lack its history so fascinating.
- orbitingpluto 11y agoWell remember Euclid's geometry is fundamentally flawed. The parallel postulate, that fact that parallel lines never meet, can be be used to really mess with a grade 7-12 math teacher's head if you invest the time to teach a couple of tricks to your kid. The simplest example is to show that an obtuse angle and a right angle are the same through a construction. The placement of points is crucial to make that work.
- catpolice 11y agoIt 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...
- javajosh 11y agoHa, that's a good one, actually. Quite elegant.
- mixedmath 11y agoYour fact probably doesn't have a name. But a good statement would be that a quadratic polynomial has at most two roots. To see this, suppose that the two circles are (x-a)^2 + (y-b)^2 = r and (x-c)^2 + (y-d)^2 = R. Subtracting gives an equation of the form (linear function in x and y) = r - R. This means that y is a linear function of x, and so we can use this to substitute in the equation for the first circle to get something of the form (quadratic function in x) = r. Then as there are at most 2 solutions for x, and each gives the corresponding solution for y, we see that two distinct circles intersect in at most 2 points.
- jschulenklopper 11y agoFrom 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?
- rspeer 11y agoThe article is missing something -- a mathematically accurate description. Not you.
- pervycreeper 11y agoI have no idea what this means, but can't imagine how it could be significant as a theorem, assuming the correct statement is reasonably similar.
- hydroo 11y agoThe theorem as stated in the article is definitely wrong. Equal length would make it correct, I think.
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- sobbybutter 11y agoNope, you can have two equal lines extending from a non-central point. Connecting the points would give you an isosceles triangle.
- jschulenklopper 11y ago> 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 :-)
- randlet 11y agoKudos to the teacher who helped her develop her idea instead of scolding her for using ideas that weren't being taught in his classroom.
- adt2bt 11y agoHow common is that?
- noobie 11y agoscolding [a student] for using ideas that weren't being taught in his classroom.? Very common!
- randlet 11y agoPretty common (in North America anyways) I think. In speaking with teacher friends my impression is that math education has become quite formulaic with a large focus on learning algorithms to solve problems that occur in standardised tests.
- fche 11y ago(that's not the same as scolding being common)
- theideasmith 11y agoI happen to be in 10th grade in the USA and really love math - and using it to solve applied problems - but the math taught in school is formulaic and mechanical. We aren't encouraged when we start asking questions not covered by the curriculum or congratulated when we answer them ourselves. It would be really nice if we had more teachers like the one in this article. I'm sure many more articles would be written then.
- PhantomGremlin 11y agoI loved the brief summary at the top. In contrast, many times there are "long form" articles which expect me to invest 10 minutes reading them before I even have a good idea what they're about. You know what I mean: Someone grew up privileged, or in the 'hood. Then had a plethora of tangential life experiences. Then maybe an epiphany. Then we begin to read something about the purported topic.
- Jugurtha 11y agoThat was my thought, exactly. I'm rather into Bottom Line Up-Front and that website seems to provide a tight summary/outline for every story. Writing well is something I'm shooting for because I'm always all over the place and I need more discipline.
- yxitcti 11y agoIsn't this common knowledge? Ask any drafter ever and they'll tell you it is. It seems pretty generous to say she "discovered a new theorem."
- mankash666 11y agoSeems 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.
- sound_of_basker 11y agoSeriously. I recall proving that there is one and only one circle that can be drawn through 3 distinct points on a Euclidean plane. This "theorem" can be thought of as a corollary of that.
- hasenj 11y agoUntil 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.
- dkopi 11y agoA circle is defined as all the points in the same distance from a certain center point. But a point thats distanced from the circle circumference by R isn't necessarily the circles center. But if you can draw 3 (and hence more) lines from a point to the circles circumference that are all the same lenght, that is the circles center, and the distance is the radius.
- Houshalter 11y agoI actually used this idea once. I had three known points which were approximately equidistant from an unknown center. I wanted to find the center. So I used a hillclimbing algorithm to search for the center by guessing points and seeing how close they were. The fitness function was the difference between the proposed center and the three points. The idea being to minimize the distance between their. If the lines were exactly the same length, I would have found the center. It didn't work at all though. It gave wildly incorrect answers, and sometimes even converged on infinity... Even when running it many many times to avoid local optima.
- jschulenklopper 11y ago> I had three known points which were approximately equidistant from an unknown center. I wanted to find the center. See this: http://www.mathopenref.com/const3pointcircle.html http://www.mathopenref.com/const3pointcircle.html
- Houshalter 11y ago
- dkopi 11y ago"Barbi prefers a future in theater rather than math." The gender wage gap, explained in one sentence.
- Luc 11y agoI never know what these 'grade' designations mean. Anyway, it appears to refer to 15 to 16-year-olds. https://en.wikipedia.org/wiki/Tenth_grade https://en.wikipedia.org/wiki/Tenth_grade
- aharonovich 11y agoIn Israel, first grade is for 6 year olds, getting out of kindergarten and into elementry school, so if you want grade to age conversion just add 6, ie 10th grade = 16 yo
- canucount 11y agoSo first grade is 1 + 6 = 7 by your algorithm?
- cbfhbcbd 11y agoIt's truly amazing how talented the Jewish people are. So many of our greatest minds come from that culture. I wish her my congratulations.
- commenter23 11y agoIt's a fluff piece. The real story here is that a 10th grader, after using a Theorem that wasn't taught in class, was encouraged to prove it - which she did, successfully. The teacher then sent it to a few academics who were thought that was a rather impressive accomplishment for a 10th grader, so they wrote her some encouraging words. That's it. The theorem and its proof are in Euclid's Elements, (Book 3 Proposition 9: http://aleph0.clarku.edu/~djoyce/elements/bookIII/propIII9.html http://aleph0.clarku.edu/~djoyce/elements/bookIII/propIII9.h...)
- 21 11y agoMy memories are a bit faded, but when I was in 10th grade in an EU country we did similar geometry problems. This doesn't seem considerably harder than the kinds of problems I remember doing. So I'm not sure how impressive this is, maybe a teacher should weigh in.
- db48x 11y agoIt's already fairly well established that any three points define a circle. Similarly, most circles have a center, and will therefore have radii which join those points to the center.
- jschulenklopper 11y ago> Similarly, most circles have a center. I think that all circles have a center.
- qwerty823 11y agoActually, they can't be co-linear as well.
- denzil_correa 11y agoSome Hebrew reports say that story has been exaggerated by the media. She proved a theorem from Euclid Elements in a different way than Euclid. https://www.facebook.com/MadaGB/photos/a.144320005726807.32753.144317442393730/593292164162920/?type=3&theater https://www.facebook.com/MadaGB/photos/a.144320005726807.327...
- slowmovintarget 11y agoThank you. That makes a lot more sense.
- namelezz 11y ago> "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" Isn't this the definition of a circle, collection of points with the same distance to its center?
- catwell 11y agoThat theorem, assuming "lines" means "segments of equal length", is a trivial consequence of "three non-aligned points define a unique circle", a well-known theorem taught in grade school (at least in France).
- j238 11y agoSeems like something most people who do geometry know intuitively without having to say out loud. I'm guessing she said something, the teacher questioned it. Then when the teacher realized she was right, she went overboard and made the case that a statement of the obvious constituted a "new theorem."
- natalys 11y agoBadly translated into english. I read the Hebrew. Should say something like. "If 3 or more equal lines extend from a single point to the edge of an arc, then the arc is part of a circle, the point being the center and the lines the radii"