$\left\{ \begin{gathered}
  3y + x =  - 9 \hfill \\
  ax - y = 3\,\,\,\, \hfill \\ 
\end{gathered}  \right. \to (3a + 1)x = 0 \Rightarrow x = 0\,\,,\,\,y =  - 3$
$ \Rightarrow A(0, - 3)$
$\left\{ \begin{gathered}
  3y + x =  - 9 \hfill \\
  y = x\,\,\,\,\,\,\,\,\,\,\,\,\,\, \hfill \\ 
\end{gathered}  \right. \to x - y = \frac{9}{4} \Rightarrow B( - \frac{9}{4}, - \frac{9}{4})$
$\left\{ \begin{gathered}
  ax - y = 3 \hfill \\
  y = x\,\,\,\,\,\,\,\,\,\,\,\,\,\, \hfill \\ 
\end{gathered}  \right. \to (a - 1)x = 3 \Rightarrow C(\frac{3}{{a - 1}},\frac{3}{{a - 1}})$

چون مرکز دایره محیطی روی $(y = x)BC$ قرار دارد پس $A = \widehat {90}$

1) $AB = \frac{{ - 3/4}}{{9/4}} =  - \frac{1}{3}$ شیب
2) $BC = 1$ شیب
3) $\frac{1}{{\tan C}} = \tan B = \frac{{1 + 1/3}}{{1 - 1/3}} = 2$
4) $\tan (B - C) = \frac{{\tan B - \tan C}}{{1 + \tan B\tan C}} = \frac{{2 - 1/2}}{2} = \frac{3}{4}$