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| Soroban Quote | Reply | This message was updated on 9/12/2002 3:05:11 PM by Soroban | "Triangular" problems posted on: 9/12/2002 3:01:41 PM These are "oldies", but someone may enjoy them. 1) On the xy-plane, is there an equilateral triangle whose vertices have integer coordinates? Find one ~ or prove it impossible. 2) In 3-dimensional space, is there a regular tetrahedron whose vertices have integer coordinates? Find one ~ or prove it impossible |
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Mr. Fox
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"Triangular" problems
replied on: 3/1/2004 3:08:18 PM quote:Taking the simplest scenario with on side of the triangle parallel to the x axis and the equation for the height (x) of an equilateral trialngle with sides (a) x=a/2(3)^-2 a must always be even to ensure that the x coordinate of the unknown vertex is an integer. But an even number divided by 2 is always an integer. And an integer multiplied by the square root of three (a non-terminating number) will never be an integer. So this scenario is impossible. Given that there are an infinite number of other orientations of the triangle so that two of its vertices have integral coordinates, I don't know how to prove that none of them have an integral vertex. I suspect that you could use fancy math to find an example that comes very, very close to three integral verteces. But that 3^-2 will always prevent it from being spot on. I haven't worked on the tetrahedron problem yet |
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Mr. Fox
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"Triangular" problems
replied on: 3/1/2004 5:25:29 PM quote:If an equilateral triangle doesn't have a solution then a regular tetrahedron made up of 4 equilateral triangles shouldn't work either and for the same reason. |
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