Let's say the three vertices of the triangle are
$$
\vec{v}_1 = \left [ \begin{matrix} x_1 \\ y_1 \\ z_1 \end{matrix} \right ],
\quad
\vec{v}_2 = \left [ \begin{matrix} x_2 \\ y_2 \\ z_2 \end{matrix} \right ],
\quad
\vec{v}_3 = \left [ \begin{matrix} x_3 \\ y_3 \\ z_3 \end{matrix} \right ]
$$
You can construct a two-dimensional coordinate system, where $\vec{v}_1$ is at origin, $\vec{v}_2$ is on the positive $x$ axis, and $\vec{v}_3$ is somewhere above the $x$ axis (i.e., has a positive $y$ coordinate).
If we use
$$\vec{V}_1 = \left [ \begin{matrix} 0 \\ 0 \end{matrix} \right ], \quad
\vec{V}_2 = \left [ \begin{matrix} h \\ 0 \end{matrix} \right ], \quad
\vec{V}_3 = \left [ \begin{matrix} i \\ j \end{matrix} \right ]$$
then the rules to keep the edge lengths intact are
$$\left\lbrace\begin{aligned}
h^2 &= \lVert \vec{v}_2 - \vec{v}_1 \rVert^2 \\
i^2 + j^2 &= \lVert \vec{v}_3 - \vec{v}_1 \rVert^2 \\
(h-i)^2 + j^2 &= \lVert \vec{v}_3 - \vec{v}_2 \rVert^2
\end{aligned}\right.$$
i.e.
$$\left\lbrace\begin{aligned}
h^2 &= (x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2 \\
i^2 + j^2 &= (x_3 - x_1)^2 + (y_3 - y_1)^2 + (z_3 - z_1)^2 \\
(h-i)^2 + j^2 &= (x_3 - x_2)^2 + (y_3 - y_2)^2 + (z_3 - z_2)^2
\end{aligned}\right.$$
If the three points are all separate and not on the same line, there is a solution:
$$\begin{aligned}
h &= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \\
i &= \frac{(x_3 - x_1)(x_2 - x_1) + (y_3 - y_1)(y_2 - y_1) + (z_3 - z_1)(z_2 - z_1)}{h} \\
j &= \sqrt{(x_3 - x_1)^2 + (y_3 - y_1)^2 + (z_3 - z_1)^2 - i^2}
\end{aligned}$$
This means that in the triangle, if we drop a line from the third vertex, perpendicular to the line between the first two vertices, the two lines intersect at distance $i$ from the first vertex towards the second vertex. (But note that $i$ can be negative, meaning the opposite direction.) The distance between the third vertex and the intersection is $j$.
OP has a triangle with $z_1 = z_2 = 0$, and wants to find out where the third vertex would be, if the triangle is rotated around the first edge to bring the third vertex to the $xy$ plane.
First, we need two unit vectors. The first unit vector, $\hat{u}$, is from $\vec{v}_1$ towards $\vec{v}_2$:
$$\hat{u} = \frac{\vec{v}_2 - \vec{v}_1}{\lVert\vec{v}_2 - \vec{v}_1\rVert} = \left [ \begin{matrix} \frac{x_2 - x_1}{h} \\ \frac{y_2 - y_1}{h} \\ 0 \end{matrix} \right ]$$
The second vector is perpendicular to the first, but also on the $xy$ plane. There are two options:
$$\hat{v}_{+} = \left [ \begin{matrix} \frac{y_2 - y_1}{h} \\ \frac{x_1 - x_2}{h} \\ 0 \end{matrix} \right ] \quad \text{or} \quad
\hat{v}_{-} = \left [ \begin{matrix} \frac{y_1 - y_2}{h} \\ \frac{x_2 - x_1}{h} \\ 0 \end{matrix} \right ]$$
Typically, you pick the one that is in the same halfspace as $\vec{v}_3$, i.e. has the larger (positive) dot product; this corresponds to the smaller rotation angle:
$$\hat{v} = \begin{cases}
\hat{v}_{+}, & \hat{v}_{+} \cdot \vec{v}_3 \ge \hat{v}_{-} \cdot \vec{v}_3 \\
\hat{v}_{-}, & \hat{v}_{+} \cdot \vec{v}_3 \lt \hat{v}_{-} \cdot \vec{v}_3 \end{cases}$$
Then, the location of the third vertex on the $xy$ plane is $\vec{v}_3^\prime$,
$$\vec{v}_3^\prime = \vec{v}_1 + i \hat{u} + j \hat{v}$$
This is the exact same location you get, if you rotate the triangle around the edge between vertices $\vec{v}_1$ and $\vec{v}_2$, bringing the third vertex also to the $xy$ plane. The two options, $\hat{v}_{+}$ and $\hat{v}_{-}$ correspond to rotations that differ by 180°.
