Latin bitrades, dissections of equilateral triangles and abelian groups

Let $T = (T^{\textstyle \ast}, T^{\scriptscriptstyle \triangle})$ be a spherical latin bitrade. With each $a=(a_1,a_2,a_3)\in T^{\textstyle \ast}$ associate a set of linear equations $\eq(T,a)$ of the form $b_1+b_2=b_3$, where $b = (b_1,b_2,b_3)$ runs through $T^{\textstyle \ast} \setminus \{a\}$. Assume $a_1 = 0 = a_2$ and $a_3 = 1$. Then $\eq(T,a)$ has in rational numbers a unique solution $b_i = \bar b_i$. Suppose that $\bar b_i \ne \bar c_i$ for all $b,c \in T^{\textstyle \ast}$ such that $b_i \ne c_i$ and $i \in \{1,2,3\}$. We prove that then $T^{\scriptscriptstyle \triangle}$ can be interpreted as a dissection of an equilateral triangle. We also consider group modifications of latin bitrades and show that the methods for generating the dissections can be used for a proof that $T^{\textstyle \ast}$ can be embedded into the operational table of a finite abelian group, for every spherical latin bitrade $T$.