Step 1
\documentclass[pstricks,12pt,border=15pt]{standalone}
\usepackage{pst-eucl}
\begin{document}
\begin{pspicture}[showgrid](-1,-3)(7,5)
\pstTriangle(2,4){A}(0,0){B}(6,0){C}
\pstCircleABC[PosAngle=60]{A}{B}{C}{O}
\pstHomO[HomCoef=0.6666]{A}{B}[M]% 2/3 --> 0.6666
\end{pspicture}
\end{document}
Note: HomCoef
cannot accept RPN 2 3 div
so I have to insert 0.6666
(3 decimals places should suffice, I think).

Step 2
\documentclass[pstricks,12pt,border=15pt]{standalone}
\usepackage{pst-eucl}
\begin{document}
\begin{pspicture}[showgrid](-1,-3)(7,5)
\pstTriangle(2,4){A}(0,0){B}(6,0){C}
\pstCircleABC[PosAngle=60]{A}{B}{C}{O}
\pstHomO[HomCoef=0.6666]{A}{B}[M]% 2/3 --> 0.6666
\pstRotation[AngleCoef=0.3333,RotAngle=\pstAngleAOB{C}{O}{A}]{O}{C}[M']
\end{pspicture}
\end{document}
Note: AngleCoef
must come before RotAngle
. It is not commutative!

Step 3 (Final)
\documentclass[pstricks,12pt,border=15pt]{standalone}
\usepackage{pst-eucl}
\begin{document}
\begin{pspicture}[showgrid=false](-1,-3)(7,5)
\pstTriangle(2,4){A}(0,0){B}(6,0){C}
\pstCircleABC[PosAngle=225]{A}{B}{C}{O}
\pstHomO[HomCoef=0.6666]{A}{B}[M]% 2/3 --> 0.6666
\pstRotation[AngleCoef=0.3333,RotAngle=\pstAngleAOB{C}{O}{A}]{O}{C}[M']
\psset{PointName=none,PointSymbol=none}
\pstRotation[AngleCoef=0.3333,RotAngle=\pstAngleAOB{B}{A}{C}]{A}{B}[P1]
\pstRotation[AngleCoef=0.6666,RotAngle=\pstAngleAOB{B}{A}{C}]{A}{B}[P2]
\pstInterLL{B}{C}{A}{P1}{Q1}
\pstInterLL{B}{C}{A}{P2}{Q2}
\psset{Mark=MarkHash,MarkAngle=90,MarkAngleRadius=.8}
\pstMarkAngle{B}{A}{Q1}{}
\pstMarkAngle{Q1}{A}{Q2}{}
\pstMarkAngle{Q2}{A}{C}{}
\psset{linestyle=dashed}
\psline(A)(Q1)
\psline(A)(Q2)
\end{pspicture}
\end{document}
Note: We have \pstSegmentMark
(ends with Mark
) to mark a segment but we have \pstMarkAngle
(begins with Mark
) to mark an angle. It seems the package author likes making inconsistent names.

Last Edit
\pscalculate
from pst-calculate
package can make me possible to insert infix calculations fed to the HomCoef
and AngleCoef
.
\documentclass[pstricks,12pt,border=15pt]{standalone}
\usepackage{pst-eucl}
\usepackage{pst-calculate}
\begin{document}
\begin{pspicture}[showgrid=false](-1,-3)(7,5)
\pstTriangle(2,4){A}(0,0){B}(6,0){C}
\pstCircleABC[PosAngle=225]{A}{B}{C}{O}
\pstHomO[HomCoef=\pscalculate{2/3}]{A}{B}[M]% Now without hard coded 0.6666
\pstRotation[AngleCoef=\pscalculate{1/3},RotAngle=\pstAngleAOB{C}{O}{A}]{O}{C}[M']
\psset{PointName=none,PointSymbol=none}
\pstRotation[AngleCoef=\pscalculate{1/3},RotAngle=\pstAngleAOB{B}{A}{C}]{A}{B}[P1]
\pstRotation[AngleCoef=\pscalculate{2/3},RotAngle=\pstAngleAOB{B}{A}{C}]{A}{B}[P2]
\pstInterLL{B}{C}{A}{P1}{Q1}
\pstInterLL{B}{C}{A}{P2}{Q2}
\psset{Mark=MarkHash,MarkAngle=90,MarkAngleRadius=.8}
\pstMarkAngle{B}{A}{Q1}{}
\pstMarkAngle{Q1}{A}{Q2}{}
\pstMarkAngle{Q2}{A}{C}{}
\psset{linestyle=dashed}
\psline(A)(Q1)
\psline(A)(Q2)
\end{pspicture}
\end{document}