<s>[INST] Proving that the lines cut the coordinate axes in concylic points The lines $2x +3y +19 = 0$ and $9x+6y-17 = 0$ cut the coordinate axes in concyclic points.What would be the fastest method to prove it manually?Is it possible to prove the statement without having to find the equation of the circle?
 [/INST] Your four points are 
$\left(0\mid-\frac{19}3\right);
\left(-\frac{19}2\mid 0\right);
\left(0\mid\frac{17}6\right);
\left(\frac{17}9\mid 0\right);
$    
If 
$\begin{vmatrix}
0^2+\left(-\frac{19}3\right)^2&0&-\frac{19}3&1\\
\left(-\frac{19}2\right)^2+0^2&-\frac{19}2&0&1\\
0^2+\left(\frac{17}9\right)^2&0&\frac{17}9&1\\
\left(\frac{17}9\right)^2+0^2&\frac{17}9&0&1\\
\end{vmatrix}=0$
then your four points are concyclic.    
Since the RHS is 0, the above determinant can be simplified into
$\begin{vmatrix}
19^2&0&-19·3&3^2\\
19^2&-19·2&0&2^2\\
17^2&0&17·6&6^2\\
17^2&17·9&0&9^2
\end{vmatrix}$ 
</s>