  In this paper, we discuss some dimension results for triangle sets of compact
sets in $\mathbb{R}^2$. In particular, we prove that for any compact set $F$ in
$\mathbb{R}^2$, the triangle set $\Delta(F)$ satisfies \[ \dim_{\mathrm{A}}
\Delta(F)\geq \frac{3}{2}\dim_{\mathrm{A}} F. \] If $\dim_{\mathrm{A}} F>1$
then we have \[ \dim_{\mathrm{A}} \Delta(F)\geq 1+\dim_{\mathrm{A}} F. \] If
$\dim_{\mathrm{A}} F>4/3$ then we have the following better bound, \[
\dim_{\mathrm{A}} \Delta(F)\geq \min\left\{\frac{5}{2}\dim_{\mathrm{A}}
F-1,3\right\}. \] Moreover, if $F$ satisfies a mild separation condition then
the above result holds also for the box dimensions, namely, \[
\underline{\dim_{\mathrm{B}}} F\geq \frac{3}{2}\underline{\dim_{\mathrm{B}}}
\Delta(F) \text{ and }\overline{\dim_{\mathrm{B}}} F\geq
\frac{3}{2}\overline{\dim_{\mathrm{B}}} \Delta(F). \]
