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.
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.
> 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.
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 :-)
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."
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).
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.
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.
>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.
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
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.
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.
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.
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).
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.
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.
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...)