note: for simplicity the 2-dimensional case is described here. a similar situation could be treated in higher dimensions.
liquids are distinguished by their ability to change form whilst retaining constant volume. a liquid triangle in $\mathbb{R}^2$ shares this behaviour.
let $A(t),B(t),C(t)$ be points in $\mathbb{R}^2$, we will suppress time co-ordinates in the notation, to preserve simplicity. the rate of change of $A$ with time will be denoted by $\dot A$, etc.
for any points $X,Y \in \mathbb{R}^2$ define $\mathbf{e}_{XY}$ to be a unit vector in the direction $XY$
in addition to the congruent displacements corresponding to movements describable as instantaneous translations and rotations the area $|ABC|$ of a triangle $ABC$, is also invariant under a shift of $A$ along $\mathbf{e}_{BC}$, and similarly for the other sides.
let $\alpha, \beta, \gamma, \kappa $ be scalar functions of $t,A,B,C$ and let $\mathbf{e}$ be a unit vector depending on the same variables, and set:
$$ \dot A = \alpha\; \mathbf{e}_{BC}+\kappa \mathbf{e} \\ \dot B = \beta\; \mathbf{e}_{CA}+\kappa \mathbf{e} \\ \dot C = \gamma\; \mathbf{e}_{AB}+\kappa \mathbf{e} \\ $$
Q: do these equations define a trajectory for which $|ABC|$ remains invariant? are any restrictions required on the functions defining the motion?
auxiliary question: instantaneous rotation is not catered for in this simplistic description. this would be fairly easy to remedy in the 2-D case, but how should one improve the formulation to remedy this deficiency for the general case?
I think what you're asking in general is this: Let $X_i(t)$ be the coordinates of the vertices of an $n$-simplex in $\mathbb{R}^n$. What functions can we allow for the $X_i(t)$ so that the volume is preserved. If this is general enough for you, and not TOO general, the answer is happily elegant. You need the determinant of the $n \times n$ matrix whose $i^{\text{th}}$ column is $X_i(t) - X_0(t)$ to be constant.