I have the following situation:

I have three constant points: $P\_1(a\_1, b\_1, 0), P\_2(a\_2, b\_2, 0), P\_3(a\_3, b\_3, 0)$. These points are on a cartesian 3d space and form an equilateral triangle with a side length equal to `len1`.

I also have a second equilateral triangle let's call it $T$ (in the same 3d space) with a side length equal to `len2 < len1` defined by three points $T\_1(x\_1, y\_1, z\_1), T\_2(x\_2, y\_2, z\_2), T\_3(x\_3, y\_3, z\_3)$ and a constraint: $z\_1, z\_2, z\_3 > 0$. $len2$ is known.

The last known thing are 3 Euler angles: $θ\_x, θ\_y, θ\_z$ which represent the rotation of triangle $Τ$ in the 3d space, $θ\_x$ is the rotation of triangle $T$ in the $x$ axis and so on. Constraint: $-90 < θ\_x, θ\_y, θ\_z < 90$ degrees.

The winding order of the triangle is the same, i.e. always $T\_1,T\_2,T\_3$, not possible to have $T\_3, T\_1, T\_2$ or other combinations. The minimum angle of the face normal of triangle $T$ with plane $z=0$ is $>0$ degrees. Just mentioning this to avoid situations like placing the triangle face down.

My first question is:

**For all possible values of $(x\_1, y\_1, z\_1), (x\_2, y\_2, z\_2), (x\_3, y\_3, z\_3)$, i.e. for all possible positions of triangle $T$ with side length =`len2` and $z>0$ on a given rotation of the triangle $T$: $θ\_x, θ\_y, θ\_z$, is there a unique vector $(L\_1, L\_2, L\_3)$ where $L\_1 = Dist(P\_1, T\_1), L\_2 = Dist(P\_2, T\_2), L\_3 = Dist(P\_3, T\_3)$ ?**

My second question is:

**If the answer to the previous question is yes, then given $L\_1, L\_2, L\_3$ and the rotation $θ\_x, θ\_y, θ\_z$ of triangle $Τ$, what are the equations for finding the coordinates of the three points $T\_1, T\_2, T\_3$ of triangle $T$?**

Thank you in advance.