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