$$L$$ is a variable line such that the algebraic sum of the distances of the points $$\left( {1,\,1} \right),\,\left( {2,\,0} \right)$$   and $$\left( {0,\,2} \right)$$  from the line is equal to zero. The line $$L$$ will always pass through :

Solution :
Let the line be $$y=mx+c$$    or $$mx-y+c=0.$$
The algebraic sum of the distances $$ = \frac{{m - 1 + c}}{{\sqrt {1 + {m^2}} }} + \frac{{2m + c}}{{\sqrt {1 + {m^2}} }} + \frac{{ - 2 + c}}{{\sqrt {1 + {m^2}} }} = 0$$
$$ \Rightarrow 3m + 3c = 3\,\,{\text{or }}1 = m + c$$
So, $$\left( {1,\,1} \right)$$  satisfies $$y=mx+c$$    for all $$m,\,c.$$