I are Lee Mingde and you is crazy.
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Saturday, June 25, 2005
Equilateral Triangle Vertices with Integer Cartesian CoordinatesHere's a picture of three seeds on a Weiqi (Go) board.
If we treat each seed as a vertex, we see that the seeds form a triangle that is almost equilateral. Here comes the question: Can we place the seeds on the board such that they form a perfect equilateral triangle? Of course, the seeds can only be placed on the intersections of vertical and horizontal lines. What if the game board extends infinitely in each direction, can we form an equilateral triangle of finite size?
A more formal way of phrasing the problem is this: Does there exist three points with finite integer Cartesian coordinates such that they form an equilateral triangle? The answer is no.
Although it is trivial that a line that is 30 degrees from the horizontal or vertical will never intersect two points with integer Cartesian coordinates, it is not so obvious if we can form equilateral triangles that does not have any edges that are either vertical or horizontal.
Anyway, here is my proof that there does not exist three points with finite integer Cartesian coordinates such that they form an equilateral triangle.
Proof:
Assume such a set of three points exists.
Let the set be translated and/or reflected such that one of the vertices is at the origin as shown in the diagram below. The equilateral triangle in the diagram exists if and only if the above mentioned set of three points exists because translation and reflection maintains the shape of the triangle.
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