In triangle $ABC$, the equation of median through $B$ and that of median through $C$ are respectively $x+y-2=0$ and $2x+y-3=0$. If the vertex $A$ is $(-2, 1)$, then does there exist such a unique triangle?
I think that the answer is positive. I assumed coordinates of $B$ and $C$ as $(b, 2-b)$ and $(c, 3-2c)$. Since the centroid is $(1, 1)$, thus I can get two linear equations in $b$ and $c$. This gives unique coordinates of $B$ and $C$. Is this the correct approach or is there another method?
Your method works fine. You get $b+c=5,b+2c=3$, so $b=7,c=-3$, so $B$ is $(7,-5)$ and $C$ is $(-2,7)$.
Of course, it is not the correct approach. There are usually many correct solutions!