MIT Mathematician confirms: Israeli 10th-grader discovers new geometric theorem | Hacker News Reader

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

israelhayom.com

50 pointsfforflo63 comments

It'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.h...)

My 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.

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.

The article is missing something -- a mathematically accurate description. Not you.

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.

> 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.

Ha, that indicates that I'm clearly thinking more as a programmer than as a mathematician :-) Thanks.

The definition of a circle is literally the set of all points whose distance from a point P is r, so yeah, I'm in agreement.

EDIT: in a plane

The lines being tangent to the circle would also work.

Nope, you can have two equal lines extending from a non-central point. Connecting the points would give you an isosceles triangle.

As well as hitting the edge of the circle at right angles.

I think it's important that they are talking about lines not line-segments. By definition, lines extend to infinity in both directions. But I agree, the wording is still confusing. I would have worded it more like "if three lines pass through the same point and the edge of a circle, then the point is the center of the circle and the lines lie along the radii of the circle."

I 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.

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

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?

Imagine a point inside a circle, close to the edge, and draw two lines to different points to the edge. That point isn't necessarily the center.

Unless you add a third line of equal distance that needs to go to yet another point on the edge. The only way to make that work is if the point is in the center (and the lines automatically are radii).

With only two lines of equal distance. the point may not be in the center.

Different Trump.

http://www.crainsnewyork.com/article/20150806/BLOGS02/150809...

>oh hang on, I see now: Trump Foundation

No relation.

http://www.ynetnews.com/articles/0,7340,L-4774818,00.html

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

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

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.

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

Haha, finally I come across someone saying this on the internet. But you're right it's exactly like that.

A 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.

I 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.

> 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

That looks like it would be quite difficult to program.

https://en.wikipedia.org/wiki/Circumscribed_circle#Circumcen...

>A circle is defined as all the points in the same distance from a certain center point.

I think they also need to be on the same plane, otherwise it's a hollow sphere.

Also, by that definition, there's an infinity of centers. There can be a line that passes through the center and perpendicular to the plane on which the circle lies. Every point of that line is equally distant from the points of the circle.

And for an infinity of planes parallel to our plane of interest, the intersection of the line and that plane gives the center of the projection of our first circle onto that new plane.

Of course were limiting this to a 2d plane.

I believe a plane is two-dimensional by definition. But I get where you're going. Isn't geometry fun!

Some 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.144320005726807.327...

Thank you. That makes a lot more sense.

Isn'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."

"Barbi prefers a future in theater rather than math."

The gender wage gap, explained in one sentence.

I 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

In 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

So first grade is 1 + 6 = 7 by your algorithm?

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!

I 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.

Pretty 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.

(that's not the same as scolding being common)

I 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.

That 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.

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...

Your 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.

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

> This is something you'd expect Euclid to write about

Which is what makes the lack its history so fascinating.

Well 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.

That 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).

> "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?

It'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.

> Similarly, most circles have a center.

I think that all circles have a center.

Actually, they can't be co-linear as well.

Seems 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."

Badly 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"

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.

Seriously. 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.

It's truly amazing how talented the Jewish people are. So many of our greatest minds come from that culture.

I wish her my congratulations.

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