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July 8th, 2018, 04:26 AM   #1
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Finding A Point Which Is Equidistant From Given Points

Hello my dears,

I have 13 distinct points with known coordinates.

Is it possible to find a point which is equidistant from the 13 known points?
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July 8th, 2018, 05:18 AM   #2
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Sorry, but you get NO help until you send Maschke his prize:
http://mymathforum.com/lounge/342377-competition-2.html
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July 8th, 2018, 05:24 AM   #3
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Quote:
Originally Posted by Denis View Post
Sorry, but you get NO help until you send Maschke his prize:
http://mymathforum.com/lounge/342377-competition-2.html
I said:
"The member who collects 10 POINTS, I will send a prize to him via a shipping company."
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July 8th, 2018, 06:12 AM   #4
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Quote:
Originally Posted by Hussain2629 View Post
Hello my dears,

I have 13 distinct points with known coordinates.

Is it possible to find a point which is equidistant from the 13 known points?
Yes.

Can you give me my point over PM?

-Dan
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July 8th, 2018, 07:43 AM   #5
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Quote:
Originally Posted by Hussain2629 View Post
I said:
"The member who collects 10 POINTS, I will send a prize to him via a shipping company."
BUT it is illegal to offer a prize without saying what the prize is:
poor Maschke could possibly have spent countless hours getting
to 10 points....then to receive a pack of chewing gum as reward
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July 8th, 2018, 11:16 AM   #6
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Quote:
Originally Posted by Hussain2629 View Post
Is it possible to find a point which is equidistant from the 13 known points?
Not in general, no.

At least not in two or three dimensional space. If you have enough dimensions it's always going to be possible I should think.
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July 8th, 2018, 12:38 PM   #7
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Not in general, no.

At least not in two or three dimensional space. If you have enough dimensions it's always going to be possible I should think.
Right. Just as 2 points determine a 0-sphere (a pair of points) in 1D, and 3 points determine a 1-sphere (a circle) in 2D, and 4 points determine a 2-sphere in 3D, 13 points would determine an 11-sphere living in 12D space.

https://en.wikipedia.org/wiki/N-sphere
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July 8th, 2018, 12:41 PM   #8
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Originally Posted by v8archie View Post
Not in general, no.

At least not in two or three dimensional space. If you have enough dimensions it's always going to be possible I should think.
That's pretty much what I had come up with. And even if I'm wrong about the number of dimensions used I would imagine we could construct a metric which should fix any problems left over from the dimensionality argument.

Admittedly the OP wouldn't be considering such.

-Dan
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July 8th, 2018, 12:47 PM   #9
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Actually you can do this in the plane if you can pick the metric. You can even do it on a line. The discrete metric is $d(x,y) = 1$ if $x \neq y$ and $d(x,y) = 0$ if $x = y$.

Then given any 13 points in the plane or on a line, pick any point distinct from all of them. Then its distance from each of the 13 points is 1.
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