> 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?
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 :-)
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.
> 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.
EDIT: in a plane
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?
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).
No relation.
http://www.crainsnewyork.com/article/20150806/BLOGS02/150809...