  Given an equilateral triangle with $a$ the square of its side length and a
point in its plane with $b$, $c$, $d$ the squares of the distances from the
point to the vertices of the triangle, it can be computed that $a$, $b$, $c$,
$d$ satisfy $3(a^2+b^2+c^2+d^2)=(a+b+c+d)^2$. This paper derives properties of
quadruples of nonnegative integers $(a,\, b,\, c,\, d)$, called triangle
quadruples, satisfying this equation. It is easy to verify that the operation
generating $(a,\, b,\, c,\, a+b+c-d)$ from $(a,\, b,\, c,\, d)$ preserves this
feature and that it and analogous ones for the other elements can be
represented by four matrices. We examine in detail the triangle group, the
group with these operations as generators, and completely classify the orbits
of quadruples with respect to the triangle group action. We also compute the
number of triangle quadruples generated after a certain number of operations
and approximate the number of quadruples bounded by characteristics such as the
maximal element. Finally, we prove that the triangle group is a hyperbolic
Coxeter group and derive information about the elements of triangle quadruples
by invoking Lie groups. We also generalize the problem to higher dimensions.
