The three vertices of a triangle are lattice points. The triangle contains no other lattice points but its interior contains exactly one lattice point.

Prove that the interior lattice point is the triangle's centroid.

(In reply to One Solution (spoiler) by Steve Herman)

There are TWO ways to draw a triangle whose bounded segments contain only three lattice points which are the vertices of a triangle and with only a single lattice point in its interior.

One of the two ways is, as you mentioned, an isosceles triangle [SQRT(5); SQRT(5); SQRT(2)] where the vertex which meets the two sides of equal length is at the corner of a 2x2 lattice grid area and the two other vertices are the midpoints of the non-adjacent sides serving as the endpoints of the remaining side of the triangle. As you gave as example, the lattice points of the vertices may be the Cartesian coordinates (0,0), (1,2) and (2,1) with the interior lattice point being (1,1).

The other way is a scalene triangle [1; SQRT(10); SQRT(13)] which has two of its vertices in opposite corners of a 2x3 lattice grid area and the third vertex at a midpoint of one of the grid sides of length 2. The lattice points of the vertices may be the Cartesian coordinates (0,0), (1,3), (2,3) with the interior lattice point being (1,2).

In both cases the interior point is the intersection of the triangle's medians.

Posted by Dej Mar on 2011-04-11 05:26:38