De Gua's theorem is a $3$-dimensional analog of the Pythagorean theorem: The square of the area of the diagonal face of a right-angled tetrahedron is the sum of the squares of the areas of the other three faces.

For certain tetrahedra, this provides a representation of an integer $n$ as the sum of three integer squares.

Let the tetrahedron have vertices at \begin{eqnarray} & (0,0,0)\\ & (a,0,0)\\ & (0,b,0)\\ & (0,0,c) \end{eqnarray} If $a,b,c$ are integers, at least two of which are even, then the squared areas of the three triangles incident to the origin are each integer squares, and so "represent" $n=A^2$, the diagonal-face area squared.

*Example*.
Let $a,b,c$ be $2,3,4$ respectively.

The diagonal face-area squared is \begin{eqnarray} A^2 & = & \left[ (2 \cdot 3)^2 + (3 \cdot 4)^2 + (4 \cdot 2)^2 \right] \,/\, 4\\ & = & (36 + 144 +64) \,/\, 4 \\ & = & 9 + 36 + 16\\ & = & 61\\ A & = & \sqrt{61} \;. \end{eqnarray} So here, $61$ is represented as the sum of three squares: $9+36+16$.

Let $N_T(n)$ be the number of integers $\le n$ that can be represented
as a sum of three squares derived from deGua's tetrahedron theorem,
as above. Call these *tetra-realized*.
Let $N_L(n)$ be the number of integers $\le n$ that can be represented
as a sum of three squares.
$N_L$ is determined by
Legendre's three-square theorem, which
says that $n$ is the sum of three squares except when it is
of the form $n=4^a (8 b + 7)$, $a,b \in \mathbb{N}$.

I would like to know how prevalent is tetra-realization:

. What is the ratio of $N_T(n)$ to $N_L(n)$ as $n \to \infty$?Q

I would also be interested in any characterization of the tetra-realizable $n$.