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The sad life of lattice triangles

Christian Aebi, Grant Cairns

Abstract

This paper treats triangles in the plane whose vertices lie on the integer lattice, i.e., the vertices have integer coordinates. It shows that apart from trivial examples, the circumcenter, centroid and orthocenter of such triangles never all lie on the integer lattice. Several further observations are made concerning the circumcenter, centroid and orthocenter.

The sad life of lattice triangles

Abstract

This paper treats triangles in the plane whose vertices lie on the integer lattice, i.e., the vertices have integer coordinates. It shows that apart from trivial examples, the circumcenter, centroid and orthocenter of such triangles never all lie on the integer lattice. Several further observations are made concerning the circumcenter, centroid and orthocenter.
Paper Structure (4 theorems, 9 equations, 4 figures)

This paper contains 4 theorems, 9 equations, 4 figures.

Key Result

Theorem 1

If for the lattice triangle with vertices $O,(x_1,y_1),(x_2,y_2)$, the circumcenter and the centroid are both lattice points, then $\gcd(x_1,y_1,x_2,y_2)$ is a multiple of $3$.

Figures (4)

  • Figure 1: A lattice triangle with Euler line $FGH$, where $H$ is a lattice point but $F$ and $G$ are not.
  • Figure 2: A lattice triangle for which $F,G, H$ are lattice points.
  • Figure 3: A lattice triangle whose Euler line contains no lattice points.
  • Figure 4: A lattice triangle whose circumcenter $F$ and orthocenter $H$ are lattice points, but whose centroid $G$ is not.

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • proof
  • Theorem 3
  • proof