Prove that there are infinitely many different triangles in coordinate plane satisfying: 1) their vertices are lattice points 2) their side lengths are consecutive integers Remark: Triangles that can be obtained by rotation or translation or shifting the coordinate system are considered as equal triangles
Problem
Source: Latvian TST for Baltic Way 2020 P14
Tags: Number theory problems, lattice points, number theory
knightime1010
18.10.2020 19:15
There are infinitely many $3$-$4$-$5$ triangles which can be drawn with their vertices on lattice points...
Blastoor
22.10.2020 17:42
knightime1010 wrote: There are infinitely many $3$-$4$-$5$ triangles which can be drawn with their vertices on lattice points... Unfortunately, the problem asks for the triangle's sides to be consecutive integers, but any bigger $3-4-5$ triangles don't provide that.
knightime1010
22.10.2020 17:56
Blastoor wrote: knightime1010 wrote: There are infinitely many $3$-$4$-$5$ triangles which can be drawn with their vertices on lattice points... Unfortunately, the problem asks for the triangle's sides to be consecutive integers, but any bigger $3-4-5$ triangles don't provide that. Yes, but you can simply translate and/or rotate one $3$-$4$-$5$ triangle to a different position on the grid, and there are infinitely many possible positions.
emsilverfox
22.10.2020 17:58
Take the triple $\{ (0,3+x),(0,x),(4,x)\}$
Kimchiks926
22.10.2020 19:57
There was translation mistake in previous problem's formulation. It's fixed now.
IAmTheHazard
22.10.2020 20:09
This has been studied before: https://en.wikipedia.org/wiki/Integer_triangle#Heronian_triangles_in_a_2D_lattice