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/
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