Sfdwhf

With the following code:

documentclass[tikz]{standalone}



usepackage{tkz-euclide,tkz-fct,amsmath}

usetkzobj{all}





begin{document}

    begin{tikzpicture}[anchor=center]

        tkzInit[xmin=-1, xmax=3, ymin=-1,ymax=3]

        tkzDefPoints{.5/2/P_1, 2.5/0/P_2, 1.5/1/M,2/1.5/A}



        tkzDrawX[noticks, label={(operatorname{Re}(x) )}]

        tkzDrawY[noticks, label={(operatorname{Im}(x) )}]



        tkzDrawPoints[fill=black, size=1mm](P_1,P_2,M)

        tkzMarkRightAngle(A,M,P_1)

        tkzFct[domain=-1:3, color=red, thick]{x-.5}

        draw (P_1) -- (P_2);

        tkzLabelPoints[above right](P_1,P_2)

        tkzLabelPoints[right](M)



        tkzText[color=black](1.5,3){(|z-z_1|leq|z-z_2| )}

    end{tikzpicture}

end{document}

I'm getting:

I wanted to add a fade like this:

but I can't have the fade to be in the right angle.

How can I get this kind of fade, fadding to white?

You may rotate the shading area to the x-axis, shade, then rotate back with 'transform canvas'

documentclass[tikz,border=5mm]{standalone}

usepackage{tkz-euclide}

usetkzobj{all}

begin{document}

begin{tikzpicture}

coordinate[label=above right:$P_1$] (P1) at (.5,2);

coordinate[label=above right:$P_2$] (P2) at (2.5,0);

coordinate[label=right:$M$] (M) at (1.5,1);

coordinate (A) at (2,1.5);

pgfmathsetmacro{a}{.5-sqrt(2)}

pgfmathsetmacro{b}{.5+sqrt(12.5)}



shade[top color=white,bottom color=red!50,

transform canvas={rotate around={45:(.5,0)}}] 

(a,0) rectangle (b,.5);

draw[-latex] (-1,0)--(3.5,0) node[below]{rm{Re}$(x)$};

draw[-latex] (0,-1)--(0,3.5) node[left]{rm{Im}$(x)$};



draw (P1) -- (P2);

draw[red,thick] plot[domain=-.5:3] (x,{x-.5});

foreach p in {P1,P2,M}

fill (p) circle(1pt);

node at (1.5,3){(|z-z_1|leq|z-z_2| )};

tkzMarkRightAngle(P1,M,A)

end{tikzpicture}

end{document}

This is in principle very simple but tkz-euclide seems to mess up things a bit. One can just use a shading angle, which can, of course, be computed by TikZ.

documentclass[tikz]{standalone}

usetikzlibrary{calc,backgrounds}

usepackage{amsmath}

DeclareMathOperator{re}{Re}

DeclareMathOperator{im}{Im}

begin{document}

  begin{tikzpicture}[anchor=center,declare function={f(x)=x-0.5;

    xmin=-1;xmax=3;}]

      draw[-latex] (-1.5,0) -- (3.5,0) node[below left]{$re z$};

      draw[-latex] (0,-1.5) -- (0,3.5) node[below left]{$im z$};;

      path foreach X/Y/L/P in {.5/2/P_1/45, 2.5/0/P_2/45, 1.5/1/M/0}

      {(X,Y) coordinate[label=P:$L$] (L)};

      begin{scope}[on background layer]

      shade let p1=({xmin},{f(xmin)}),p2=({xmax},{f(xmax)}),

      n1={atan2(y2-y1,x2-x1)} in 

       [left color=white,right color=red,middle color=white,shading angle=n1]

       (p1) -- (p2)  --  ($(p2)!2cm!-90:(p1)$) -- ($(p1)!2cm!90:(p2)$)

        ;

      end{scope}   

      draw[red,thick] plot[variable=x,domain=xmin:xmax] ({x},{f(x)});

      draw (P_1) -- (P_2);

      node[anchor=south,red] at (1.5,3) {$|z-z_1|leq|z-z_2| $};

  end{tikzpicture}

end{document}