documentclass{article}
usepackage{tkz-euclide}
usetkzobj{all}
usetikzlibrary{calc,patterns,angles,quotes,intersections}
begin{document}
noindenthrulefill
begin{center}
begin{tikzpicture}[scale=0.5,
  dot/.style 2 args={circle,inner sep=1pt,fill,label={#2},name=#1},
  dot2/.style 2 args={circle,inner sep=.6pt,draw=black, fill=white,label={#2},name=#1},
  dot3/.style 2 args={circle,inner sep=.8pt,draw=black, fill=white,label={#2},name=#1},
  extended line/.style={shorten >=-#1,shorten <=-#1},
  extended line/.default=1cm]

  draw[help lines,step=1] (-1,-1) grid (10,6);
  draw [->] (-1,0) -- (10,0) node [below left]  {$x$};
  draw [->] (0,-1) -- (0,6) node [below left] {$y$};

  node [dot={p1}{[below=1.5mm]$p_1$}] at (3,1) {};
  node [dot={p2}{[above right]$p_2$}] at (8,0) {};
  node [dot={p3}{[above=1mm]$p_3$}] at (3.5,2.5) {};
  node [dot={p4}{[above right]$p_4$}] at (6,3) {};

  coordinate (i1) at (intersection of p1--p2 and p3--p4);
  coordinate (i2) at (intersection of p1--p3 and p2--p4);
  draw [extended line=0.3cm] (p2) -- (i1) ;
  draw [extended line=0.3cm] (p2) -- (i2) ;
  draw [extended line=0.3cm] (p4) -- (i1) ;
  draw [extended line=0.3cm] (p1) -- (i2) ;
  draw [thick] (p1) -- (p2);
  draw [thick] (p2) -- (p4);
  draw [thick] (p3) -- (p4);
  draw [thick] (p3) -- (p1);
  draw [very thick,extended line=0.3cm] (i1) -- (i2) ;
  coordinate (i3) at (intersection of p2--p3 and i1--i2);
  coordinate (o) at (intersection of p2--p3 and p1--p4);
  coordinate (i4) at (intersection of i1--o and p2--p4);
  coordinate (i5) at (intersection of i2--o and p1--p2);
  coordinate (i6) at (intersection of i1--o and p1--p3);
  coordinate (i7) at (intersection of i2--o and p3--p4);
  draw [very thin,extended line=0.3cm] (p1) -- (p4) ;
  draw [very thin,extended line=0.3cm] (p2) -- (i3) ;
  draw [very thin,extended line=0.3cm] (i1) -- (i4) ;
  draw [very thin,extended line=0.3cm] (i2) -- (i5) ;
  node[dot2,label={}] at (o) {};
  node[dot2,label={}] at (i4) {};
  node[dot2,label={}] at (i5) {};
  node[dot2,label={}] at (i6) {};
  node[dot2,label={}] at (i7) {};
  node[dot3,label={[above]$i_1$}] at (i1) {};
  node[dot3,label={[label distance=0mm]176.0:$i_2$}] at (i2) {}; % <<<=====
end{tikzpicture}
end{center}
noindenthrulefill
end{document}