I'm having trouble with the associativity part of showing that $\text{Aff}(\mathbb{R}^2)$ is a group.
Recall that if $\left[ A, \overline{r} \right]$ denotes an affine transformation (where $A$ is the matrix representation of the linear transformation and $\overline{r}$ is the translation vector), then multiplication in $\text{Aff}(\mathbb{R}^2)$ is defined by $\left[ A, \overline{r} \right] \cdot \left[ B, \overline{s} \right] := [AB,\; B^{-1}\overline{r} + \overline{s} ].$
We must show that $\left(\left[ A, \overline{r} \right] \cdot \left[ B, \overline{s}\right] \right) \cdot [ C, \overline{t} ] = \left[ A, \overline{r} \right] \cdot \left([ B, \overline{s}] \cdot [ C, \overline{t}] \right)$
The expansion of the left side is $[ABC,\; C^{-1}(B^{-1}\overline{r} + \overline{s}) + \overline{t}]$. I've tried working from the other side and making them meet in the middle, but I can't seem to massage them the right way so that they're equal. What's the trick here?