The vertices of a triangle are lattice points (they have integer coordinates). There are no other lattice points on the boundary of the triangle, but there is exactly one lattice point inside the triangle. Show that it must be the centroid.
Solution
Let the three vertices A, B, C have vectors a, b, c. Then the lattice point D inside the triangle must have vector d = λa + μb + νc for some positive reals λ, μ, ν with sum 1. Suppose λ > ½. Consider the point 2d-a. Since d and a have integer coordinates, it does also, so it is a lattice point. But it can be written as (2λ-1)a + 2μb + 2νc, which has positive coordinates summing to 1, so it must lie strictly inside the triangle. But the only such lattice point is D. So 2d-a = d. Hence A and D coincide. Contradiction. Similarly, if λ = ½ then 2d-a = 2μb + 2νc, so it lies in the open segment between B and C. Contradiction. Hence λ < ½. Similarly, μ < ½ and ν < ½. But now consider the point a + b + c - 2d = (1-2λ)a + (1-2μ)b + (1-2ν)c. It has positive coordinates summing to 1 and is a lattice point. So it must lie strictly inside the triangle and hence be D. So d = (a + b + c)/3, which shows that D is the centroid.
© John Scholes
jscholes@kalva.demon.co.uk
29 Oct 2003
Last corrected/updated 29 Oct 03