
MIT Mathematician confirms: Israeli 10th-grader discovers new geometric theorem - fforflo
http://www.israelhayom.com/site/newsletter_article.php?id=32345
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commenter23
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...](http://aleph0.clarku.edu/~djoyce/elements/bookIII/propIII9.html))

~~~
21
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.

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jschulenklopper
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?

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

~~~
jschulenklopper
> 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 :-)

~~~
hasenj
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.

~~~
jschulenklopper
> 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.

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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.

~~~
dkopi
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.

~~~
Houshalter
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.

~~~
jschulenklopper
> 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
That looks like it would be quite difficult to program.

~~~
im3w1l
[https://en.wikipedia.org/wiki/Circumscribed_circle#Circumcen...](https://en.wikipedia.org/wiki/Circumscribed_circle#Circumcenter_coordinates)

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denzil_correa
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...](https://www.facebook.com/MadaGB/photos/a.144320005726807.32753.144317442393730/593292164162920/?type=3&theater)

~~~
slowmovintarget
Thank you. That makes a lot more sense.

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yxitcti
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."

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

The gender wage gap, explained in one sentence.

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Luc
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](https://en.wikipedia.org/wiki/Tenth_grade)

~~~
aharonovich
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

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

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randlet
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.

~~~
adt2bt
How common is that?

~~~
randlet
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.

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

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PhantomGremlin
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.

~~~
Jugurtha
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.

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eximius
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.

~~~
catpolice
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...

~~~
mixedmath
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.

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catwell
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).

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namelezz
> "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?

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db48x
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.

~~~
jschulenklopper
> Similarly, most circles have a center.

I think that _all_ circles have a center.

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j238
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."

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natalys
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"

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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.

~~~
sound_of_basker
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.

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cbfhbcbd
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.

