The following code is my attempt to find M and N such that AM=CN=BC.

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

usepackage{pst-eucl,pst-calculate}

begin{document}

defx{pscalculate{6-3.3541}}

begin{pspicture}[showgrid](-1,-1)(8,8)

pstGeonode[PosAngle={180,90,0,-90}](1,2){B}(2.5,5){C}(6,2){A}(x,2){M}

pspolygon(B)(C)(A)

pstRotation[PosAngle=90,RotAngle=75.9638]{C}{B}[N]



psGetDistanceAB[xShift=-8,yShift=4](B)(C){M}

psGetAngleABC[AngleValue=true,MarkAngleRadius=.5,LabelSep=0.5,%

ShowWedge=false,xShift=-5,yShift=7,arrows=->](B)(C)(A){}



end{pspicture}

end{document}

However, I am trying not to use psGetDistanceAB and psGetAngleABC.

Question

How to get M and N such that AM=CN=BC only with a compass and a straight edge?

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

usepackage{pst-eucl,pst-calculate,textcomp}

begin{document}

    begin{pspicture}[showgrid](-1,-1)(8,8)

    pstTriangle[PosAngle={180,90,0}](1,2){B}(2.5,5){C}(6,2){A}

    pstInterLC[PointSymbolB=none,PointName=none]{A}{C}{C}{B}{N}{N0}uput[0](N){N}

    pstInterLC[PointSymbolB=none,PointName=none,

                Radius=pstDistCalc{sqrt((2.5-1)^2+(5-2)^2)}]{B}{A}{A}{}{M}{M0}uput[-90](M){M}

    psarc[origin={C}]{->}(C){0.5}{(B)}{(N)}uput{5mm}[-10](C){75.9636textdegree}

    end{pspicture}

end{document}

The shorter the code the more beautiful it becomes.

documentclass[pstricks,12pt]{standalone}

usepackage{pst-eucl}



begin{document}

begin{pspicture}(0,1)(7,6)

pstTriangle[PosAngle={180,90,0}](1,2){B}(2.5,5){C}(6,2){A}

pstInterLC[PointSymbolB=none,PointNameB=,PosAngle=45]{A}{C}{C}{B}{N}{x}

pstTranslation[PointName=none,PointSymbol=none]{B}{A}{C}[C']

pstInterLC[PointNameA=,PointSymbolA=none,PosAngleB=-90]{A}{B}{A}{C'}{y}{M}

pstMarkAngle[arrows=->,MarkAngleRadius=1.1,LabelSep=.8]{B}{C}{A}{scriptsize$75.96^circ$}

end{pspicture}

end{document}

Notes

PointNameA and PointNameB cannot be assigned none but empty. It looks like a bug.

An alternative tikz version:

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

usetikzlibrary{calc,bending,arrows.meta,quotes,angles}

usepackage{textcomp}

begin{document}



begin{tikzpicture}[dot/.style={fill,circle,inner sep=1.2pt},>= Stealth]footnotesize

pgfmathsetmacroBC{sqrt(1.5*1.5+3*3)}

pgfmathsetmacroAC{sqrt(3.5*3.5+3*3)}

draw (1,2)coordinate[dot,label=left:$B$](b)--(2.5,5)coordinate[dot,label=above:$C$](c)--(6,2)coordinate[dot,label=right:$A$](a)--cycle;

node at ([xshift=-BC cm]a) [dot,label=below:$M$] {};

node at ($(c)!BC/AC!(a)$) [dot,label=right:$N$] {};

pic["75.96textdegree",draw,->,angle eccentricity=.7,angle radius=1cm] {angle=b--c--a};

end{tikzpicture}



end{document}