Latin bitrades, dissections of equilateral triangles and abelian groups

Let $T=(T^{\ast },T^{\mathrm{△}})$ be a spherical latin bitrade. With each $a=(a_1,a_2,a_3)\in T^{\ast }$ associate a set of linear equations $\text{eq}(T,a)$ of the form $b_1+b_2=b_3$, where $b=(b_1,b_2,b_3)$ runs through $T^{\ast }\setminus \{a\}$. Assume $a_1=0=a_2$ and $a_3=1$. Then $\text{eq}(T,a)$ has in rational numbers a unique solution $b_i={\bar{b}}_i$. Suppose that ${\bar{b}}_i\mathrm{\ne }{\bar{c}}_i$ for all $b,c \in T^{\textstyle \ast}$suchthat$b_i \ne c_i$and$i \in {1,2,3}$. We prove that then $T^{\mathrm{△}}$ 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 $T^{\ast }$ can be embedded into the operational table of a finite abelian group, for every spherical latin bitrade $T$.