The triangle with the lines $y=x,x+y=2$  and line parallel to x axis through $P(h,k)$   has area  $4h^2.$ Then locus of  $P(h,k)$  represents 

$y=x,x+y=2,y=k$  are sides
Vertices are  $(1,1),(2-k,k),(k,k)$
Area  $=\frac{1}{2}\left|\begin{array}{ccc}1& 1& 1\\ 1& 2-k& k\\ 1& k& k\end{array}\right|=\frac{1}{2}(2k^2-4k+2)={(k-1)}^2$
${(k-1)}^2=4h^2\Rightarrow 2x=\pm (y-1)\Rightarrow 2lines$