A Wee Triangle

Area of $\triangle ABC$ with coordinates $(x_1,y_1),(x_2,y_2)$ and $(x_3,y_3)$ is, $\begin{aligned} \frac{1}{2} \begin{vmatrix} x_1&y_1&1 \\x_2&y_2&1 \\ x_3&y_3&1\end{vmatrix}&=\frac{1}{2} \begin{vmatrix} 0&0&1 \\2+a&2-a&1 \\ 2a&3-a&1\end{vmatrix} \\ \\ &=\frac{1}{2}\left[(2+a)(3-a)-(2-a)(2a)\right] \\ \\ &=\frac{1}{2}\left[a^2-3a+6\right ] \\ \\ &=\frac{1}{2}\left[\left(a-\frac{3}{2}\right)^2+\frac{15}{4}\right] \end{aligned}$ The above expression which represents the area of the triangle is minimized when $a=\dfrac{3}{2}$ and hence minimum the area is $\boxed{\dfrac{15}{8}}$

Therefore, $p=15,\; q=8\implies p+q=\boxed{23}$