documentclass[12pt,border=5pt]{standalone}

usepackage{newcent,pstricks,pst-eucl,amsmath,amssymb} 

usepackage{auto-pst-pdf}

begin{document}



begin{pspicture}

psset{PointSymbol=o}

pstGeonode[PointName=none,PointSymbol=none](2,4){A}(0,0){B}(6,0){C}

pstCircleABC[PosAngle=180]{A}{B}{C}{O}

pstGeonode[PosAngle=60](10,2){M}

pstMiddleAB[PointName=none,PointSymbol=none]{O}{M}{M'}

pstInterCC[DiameterB=pstDistAB{O}{M},PosAngleA=45,PosAngleB=-45]{O}{A}{M'}{}{D}{E}

{psset{nodesepB=-1}  pstLineAB{M}{D} pstLineAB{M}{E}}

ncLine{O}{D} ncLine{O}{E}

pstInterLL[PointSymbol=none,PosAngle=50,PointNameSep=.55]{O}{M}{D}{E}{H}

pstRightAngle[RightAngleSize=.2]{D}{H}{M}

ncLine{O}{M} ncLine{D}{E}

end{pspicture}

end{document}

It produces the following.

But I want to have the following result.

documentclass[12pt,border=15pt,pstricks]{standalone}

usepackage{pst-eucl} 



begin{document}

begin{pspicture}[showgrid,PointSymbol=none](-1,-3)(11,5)

pstGeonode[PointName=none](2,4){A}(0,0){B}(6,0){C}

pstCircleABC[PosAngle=180]{A}{B}{C}{O}

pstGeonode[PosAngle=60](10,2){M}

pstMiddleAB[PointName=none]{O}{M}{M'}

pstInterCC[DiameterB=pstDistAB{O}{M},PosAngleA=45,PosAngleB=-45]{O}{A}{M'}{}{D}{E}

{psset{nodesepB=-1} pstLineAB{M}{D}pstLineAB{M}{E}}

ncLine{O}{D}ncLine{O}{E}

pstInterLL[PosAngle=50,PointNameSep=.55]{O}{M}{D}{E}{H}

pstRightAngle[RightAngleSize=.2]{D}{H}{M}

ncLine{O}{M} 

ncLine{D}{E}

psset{fillcolor=white,fillstyle=solid}

pscircle(D){3pt}

pscircle(E){3pt}

pscircle(M){3pt}

end{pspicture}

end{document}

Simpler version

documentclass[12pt,border=15pt,pstricks]{standalone}

usepackage{pst-eucl,pstricks-add}



defr{pscalculate{sqrt(10)}}

begin{document}

begin{pspicture}[showgrid,PointSymbol=none](12,8)

    pstGeonode[PosAngle={180,90}](4,4){O}(11,5){M}

    pscircle(O){r}

    psCircleTangents(M)(O){r}

    pstGeonode[PosAngle={45,-45}](CircleT1){D}(CircleT2){E}

    pstMiddleAB[PosAngle=-45]{D}{E}{H}

    pstRightAngle[RightAngleSize=.2]{D}{H}{M}

    pspolygon(O)(D)(M)(E)

    psline{-o}(O)(M)

    psline{o-o}(D)(E)

end{pspicture}

end{document}

documentclass[12pt,border=15pt,pstricks]{standalone}

usepackage{pst-eucl} 



begin{document}

begin{pspicture}[showgrid,PointSymbol=none,dotsize=0.2](-1,-3)(11,5)

    pstGeonode[PointName={none,none,none,M}](2,4){A}(0,0){B}(6,0){C}(10,2){M}

    pstCircleABC[PosAngle=180]{A}{B}{C}{O}

    pstMiddleAB[PointName=none]{O}{M}{M'}

    pstInterCC[DiameterB=pstDistAB{O}{M},PosAngleA=45,PosAngleB=-45]{O}{A}{M'}{}{D}{E}

    ncLine{O}{M}    ncline[nodesepB=-1]{M}{D} ncline[nodesepB=-1]{o-}{M}{E}

    psline(D)(O)(E) ncLine{o-o}{D}{E} 

    pstInterLL[PosAngle=50,PointNameSep=.55]{O}{M}{D}{E}{H}

    pstRightAngle[RightAngleSize=.2]{D}{H}{M}

end{pspicture}

end{document}

For comparison: a TikZ version without tkz-euclide and the like.

documentclass[tikz,border=3.14mm]{standalone}

usetikzlibrary{calc,through}

tikzset{circle through 3 points/.style n args={3}{% https://tex.stackexchange.com/a/461180/121799

insert path={let    p1=($(#1)!0.5!(#2)$),

                    p2=($(#1)!0.5!(#3)$),

                    p3=($(#1)!0.5!(#2)!1!-90:(#2)$),

                    p4=($(#1)!0.5!(#3)!1!90:(#3)$),

                    p5=(intersection of p1--p3 and p2--p4)

                    in },

at={(p5)},

circle through= {(#1)}

}}



begin{document}

    begin{tikzpicture}[bullet/.style={circle,draw,fill=white,inner sep=1pt}]

    path (2,4) coordinate(A) (0,0) coordinate(B) (6,0) coordinate(C) (10,2)

    node[bullet,label=above:$M$](M){};

    node[circle through 3 points={A}{B}{C},draw] (c){};

    draw[shorten <=-1cm] 

    (tangent cs:node=c,point={(M)},solution=1) node[bullet,label=above:$D$](D){}

    -- (M) ;

    draw[shorten <=-1cm] (tangent cs:node=c,point={(M)},solution=2) 

       node[bullet,label=below:$E$] (E){} -- (M);

    draw (D) -- (c.center) node[bullet,label=left:$O$] (O){} -- (E) -- (D);

    draw (O) -- (M);

    path (intersection cs:first line={(D)--(E)}, second line={(O)--(M)})

    node[bullet,label=above right:$H$] (H){};

    draw ($(H)!2mm!(M)$) coordinate(aux) --  ($(aux)!2mm!-90:(M)$)

    -- ($(H)!2mm!(E)$); 

end{tikzpicture}

end{document}