# What is the side length of the smallest sized equilateral triangle that can be placed on the (x,y) plane where all coordinates are integers AND no side is horizontal and no side is vertical?

##### 1 Answer

There is no such triangle, due to

#### Explanation:

Without loss of generality, one of the vertices is at

Let the vertex anticlockwise from

The midpoint of the corresponding side is

The line through

The third vertex of the triangle lies on the line through the midpoint

In fact it will lie at the point:

#(m/2, n/2) + sqrt(3)/2 (-n, m) = ((m-sqrt(3)n)/2, (n+sqrt(3)m)/2)#

since the height of an equilateral triangle is

So we require

Since