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### 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,

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/
```
Associated Topics:
College Triangles and Other Polygons
High School Triangles and Other Polygons

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