Given a triangle on a plane
T
and an unknown evolution function
E(t,v1,v2,v3) -> p
where v1,v2,v3 are the normalized area coordinates of p at the current time t[<=0]; E(t,v1,v2,v3) is a system of driven harmonic oscillators.
The only operations available are
V1(t,s) -> x1, V2(t,s) -> x2, V3(t,s) -> x3
each of these starts at one of the vertices and returns the arithmetic mean of all the points along a line that intersects the opposite edge at s for the current time.
Given this information, what I would like to know is if it is possible or impossible to derive the actual value of 'p' at the current time or at least approach it over successive iterations of time.
To repeat, the problem is to approximate $E(t, v_1, v_2, v_3)$ where $t$ is time, and $v_1$, $v_2$ and $v_3$ are the normalized area coordinates (barycentric coordinates) of a point within the triangular domain, when only the three line integrals $I_1$, $I_2$, $I_3$ over $E$ across the triangle crossing over the point $(v_1, v_2, v_3)$ and one of the three vertices are known.
Note that the arithmetic mean in the question is just the line integral divided by the line length, and the line length is known when the (relative) lengths of the triangle and the vertex angles are known.
This method is usually called tomographic reconstruction, and the core mathematical concept you need is Radon transform.
In theory (projection-slice theorem), an infinite number of lines is needed at each point in time to recover the exact $E(t)$ (within each point inside the triangular domain); but a large number of lines suffices for practical approximation. The exact number of lines needed depends on $E(t)$, and is usually experimentally decided for each case.
Note, however, that the Radon transform does not reveal $E(t, v_1, v_2, v_3)$ at a specific point $(v_1, v_2, v_3)$ at time $t$: it takes the information obtained from a large number of those lines, and "reconstructs" the entire triangular domain in one transform.
Therefore, the answer is "Yes, you can reconstruct $E(t)$ to arbitrary precision by using enough lines." However, if you are only interested in one specific point $(v_1, v_2, v_3)$ (at different times $t$), you will have to do basically just as many math operations (work) to obtain the approximate value at that point as when obtaining the approximate value at all points within the triangular domain.
I am not personally familiar with Radon transform (I am aware of the practical uses of tomographic reconstruction, but haven't done any of the math myself), so unfortunately I cannot really help with the mathematical details on how to implement the transform, but the above links to Wikipedia articles should get you started; and the references in those articles should help you with the actual implementation.