Denote the distances $AB, BC, AC$ as $p,q,r$ respectively. Let's work in $\mathbb R^n$, with the Euclidean Metric.
Place $A$ at the origin $(0,0,0,\cdots,0)$, without loss of generality. Place $B$ on the positive $x_1$-axis at the known distance from $A$.So, it's coordinates are $ (p,0, \cdots ,0)$
Let the coordinates of $C$ be $(x_1,x_2 \cdots, x_n)$. Then we have the equations $\sum_{i=1}^n x_i^2=r^2, \text{ and } (x_1- p)^2+\sum_{i=2}^nx_i^2=q^2,$ from the distances we know and the usual euclidean distance formula in $\mathbb R^n$.
Some algebra reduces this to the single equation $(x_1- p)^2 +r^2-x_1^2=q^2$ and so, $x_1 = \frac{p^2+q^2-r^2}{2p}$.
In $2$-dimensions, you get two values for the y-coordinate, one above the $x$-axis and one below it. But in higher dimensions, after substituting the value of $x_1$ (That we just determined) into $\sum_{i=1}^n x_i^2=r^2$, this yields the locus of all possible locations for the third point $C$. After this, all translations, rotations and reflections( Which are represented by invertible$n \times n$ matrices), yields all possible embeddings (Or placements) of the triangle in $\mathbb R ^n$.