.

$L_1:x-y=0$

$L_2:x+2y=3$

$L_3:2x+y=6$

On solving $L_1$ and $L_2:y=1$ and $x=1$

On solving $L_1$ and $L_3$, we get

$x=2$

$y=2$

$L_2$ and $L_3$:

$x+2y=3$

$2x+y=6$

On solving $L_2 \& L_3$, we get

$x=3$

$y=0$

$AC=\sqrt{4+1}=\sqrt{5}$

$BC=\sqrt{4+1}=\sqrt{5}$

$AB=\sqrt{1+1}=\sqrt{2}$

so it is an isosceles triangle