## Pairwise distances between points on the edges of a triangle with noisy edge lengths

I select three points on a two-dimensional plane ($A$, $B$, $C$) to define the vertices of a triangle with edge lengths $L_{AB}$, $L_{BC}$, and $L_{CA}$. Each edge length has a corresponding uncertainty $U_{AB}$, $U_{BC}$, and $U_{CA}$ s.t. exact length of $L_{AB}$ is known to fall in the interval $[(L_{AB}-U_{AB}),(L_{AB}+U_{AB})]$, and so on for the other edges.

I then select three additional points, one along each edge of the triangle:

The first point, $p_1$, is placed on the edge of length $L_{AB}$ some fraction $k_1$ of the distance along the interval between vertices $A$ and $B$. Here, the distance between $p_1$ and the vertex $A$ can be defined as $Dist(p_1 -> A) = (k_1*L_{AB})$.

The second point, $p_2$, is placed on the edge of length $L_{BC}$ some fraction $k_2$ of the distance along the interval between vertices $B$ and $C$ s.t. $Dist(p_2 -> B) = (k_2*L_{BC})$.

Similarly, the third point, $p_3$, is placed on the edge of length $L_{CA}$ some fraction $k_3$ of the distance along the interval between vertices $C$ and $A$ s.t. $Dist(p_3 -> C) = (k_3*L_{CA})$.

Let $L_{(p_1,p_2)}$, $L_{(p_2,p_3)}$, and $L_{(p_3,p_1)}$ represent the set of distances between all pairs of these three points. What are their corresponding uncertainties?

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