Triangle Centers at Lattice Points

Date: 09/03/2002 at 10:58:20
From: Roger L. Guffey
Subject: Centers of a triangle

Is there a triangle that can be plotted on a rectangular grid so 
that all of its vertices and all four centers are lattice points? If 
so, what are the coordinates of the vertices, please?

Date: 09/03/2002 at 16:00:34
From: Doctor Floor
Subject: Re: Centers of a triangle

Hi, Roger,

Thanks for your question.

Yes, there is such a triangle. 

In a right triangle, the vertex of the right angle is the same point 
as the orthocenter, and the midpoint of the hypotenuse is the
circumcenter. So that is an easy start. Furthermore, the centroid is 
the sum of the vertex coordinates divided by three.

So if the vertices are taken with the right angle sides following the 
lattice, it is easy to get the centroid, orthocenter, and circumcenter 
as lattice points.

The radius of the incircle of a 3,4,5 triangle is also very
interesting. See:

   Radius of a Circle Inscribed in a Triangle
   http://mathforum.com/library/drmath/view/55138.html 

That should be sufficient for your to find the triangle you need. If
you have more questions, just write back.

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/