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The Aha!" problems 4. Six integral points


Best Answer k-man, 09 July 2014 - 05:44 PM

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#1 bonanova

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Posted 27 April 2014 - 07:48 AM

Determine the coordinates of six points on the plane with the following properties.

  1. No three points are collinear.
  2. Every pairwise distance is an integer.

You may use sketchpad, compass, ruler, straight edge, whatever you think may be useful.

The answer will be six pairs of coordinates: (xi , yi).


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#2 Perhaps check it again

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Posted 27 April 2014 - 06:25 PM

There is an example with a set of seven coordinates on the plane such that no three points are
collinear and every pairwise distance is an integer.  So, if you want a set of six coordinates, just
omit one of the seven.  The example is at the top of the second page:

 


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#3 bonanova

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Posted 03 May 2014 - 01:00 PM

Some of the considerations discussed in that paper bear on the problem at hand.

But in the spirit of the Aha! theme, let's rule out exhaustive computer constructions and go with compass and straightedge.


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#4 bonanova

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Posted 06 May 2014 - 11:40 AM

Spoiler for clue

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#5 plasmid

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Posted 07 May 2014 - 04:14 PM

I don't think you could start with the points of a 3 4 5 triangle and add three more points to make this work.
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#6 plainglazed

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Posted 07 May 2014 - 08:44 PM

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#7 DeGe

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Posted 07 May 2014 - 09:27 PM

Spoiler for could work

Edited by DeGe, 07 May 2014 - 09:29 PM.

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#8 bonanova

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Posted 09 July 2014 - 06:41 AM

Spoiler for could work

 

DeGe, sorry for the delay in answering.

 

I think your right triangle A1B1C2 will have sides of 4, 6 and sqrt(52).

 

That is, A1C2 will be irrational. But did I get your construction right?


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#9 k-man

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Posted 09 July 2014 - 05:44 PM   Best Answer

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#10 bonanova

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Posted 09 July 2014 - 05:58 PM

Here is the solution I had in mind, and k-man found it.

 

Spoiler for Start with 3 4 5 triangle

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