Prove that no three points with integer coordinates can be the vertices of an equilateral triangle.
Problem
Source:
Tags: analytic geometry, geometry proposed, geometry
WakeUp
30.10.2010 22:46
It is a classic, with quite a few generalisations. See these links: http://www.artofproblemsolving.com/Forum/viewtopic.php?t=150961 http://www.artofproblemsolving.com/Forum/viewtopic.php?t=10569 http://www.artofproblemsolving.com/Forum/viewtopic.php?f=151&t=217683 http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=237223 http://www.artofproblemsolving.com/Forum/viewtopic.php?f=57&t=241573
AopsUser101
09.02.2020 21:28
We know that using shoelace's theorem, the area will be rational. However, using the formula for the area of the equilateral triangle in terms of side length, we have that the area is irrational. Contradiction. EDIT: This is on the assumption we're on a 2D plane. Are we?
Math_legendno12
01.07.2023 13:09
To elaborate on a part of the comment above, the formula is sqrt(3)/4 x s^2. Note that a square of a surd, the side length or the distance between 2 points, is always an integer. The area is irrational, thus contradiction