documentclass[12pt,border=5pt]standalone
usepackagenewcent,pstricks,pst-eucl,amsmath,amssymb 
usepackageauto-pst-pdf
begindocument

beginpspicture
pssetPointSymbol=o
pstGeonode[PointName=none,PointSymbol=none](2,4)A(0,0)B(6,0)C
pstCircleABC[PosAngle=180]ABCO
pstGeonode[PosAngle=60](10,2)M
pstMiddleAB[PointName=none,PointSymbol=none]OMM'
pstInterCC[DiameterB=pstDistABOM,PosAngleA=45,PosAngleB=-45]OAM'DE
pssetnodesepB=-1 pstLineABMD pstLineABME
ncLineOD ncLineOE
pstInterLL[PointSymbol=none,PosAngle=50,PointNameSep=.55]OMDEH
pstRightAngle[RightAngleSize=.2]DHM
ncLineOM ncLineDE
endpspicture
enddocument

It produces the following.

But I want to have the following result.

documentclass[12pt,border=15pt,pstricks]standalone
usepackagepst-eucl 

begindocument
beginpspicture[showgrid,PointSymbol=none](-1,-3)(11,5)
pstGeonode[PointName=none](2,4)A(0,0)B(6,0)C
pstCircleABC[PosAngle=180]ABCO
pstGeonode[PosAngle=60](10,2)M
pstMiddleAB[PointName=none]OMM'
pstInterCC[DiameterB=pstDistABOM,PosAngleA=45,PosAngleB=-45]OAM'DE
pssetnodesepB=-1 pstLineABMDpstLineABME
ncLineODncLineOE
pstInterLL[PosAngle=50,PointNameSep=.55]OMDEH
pstRightAngle[RightAngleSize=.2]DHM
ncLineOM 
ncLineDE
pssetfillcolor=white,fillstyle=solid
pscircle(D)3pt
pscircle(E)3pt
pscircle(M)3pt
endpspicture
enddocument

Simpler version

documentclass[12pt,border=15pt,pstricks]standalone
usepackagepst-eucl,pstricks-add

defrpscalculatesqrt(10)
begindocument
beginpspicture[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]DEH
 pstRightAngle[RightAngleSize=.2]DHM
 pspolygon(O)(D)(M)(E)
 psline-o(O)(M)
 pslineo-o(D)(E)
endpspicture
enddocument

documentclass[12pt,border=15pt,pstricks]standalone
usepackagepst-eucl 

begindocument
beginpspicture[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]ABCO
 pstMiddleAB[PointName=none]OMM'
 pstInterCC[DiameterB=pstDistABOM,PosAngleA=45,PosAngleB=-45]OAM'DE
 ncLineOM ncline[nodesepB=-1]MD ncline[nodesepB=-1]o-ME
 psline(D)(O)(E) ncLineo-oDE 
 pstInterLL[PosAngle=50,PointNameSep=.55]OMDEH
 pstRightAngle[RightAngleSize=.2]DHM
endpspicture
enddocument

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

documentclass[tikz,border=3.14mm]standalone
usetikzlibrarycalc,through
tikzsetcircle 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)


begindocument
 begintikzpicture[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=ABC,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) -- ($(aux)!2mm!-90:(M)$)
 -- ($(H)!2mm!(E)$); 
endtikzpicture
enddocument