\documentclass{article}
\usepackage[pdftex,active,tightpage]{preview}
\setlength\PreviewBorder{2mm}
\usepackage{tikz}
\usepackage{tkz-euclide}
\usetkzobj{all}
\usetikzlibrary{shapes, calc, decorations}
\usepackage{amsmath,amssymb}
\usepackage{helvet}
\usepackage[eulergreek]{sansmath}

\begin{document}
\begin{preview}
\begin{tikzpicture}[very thick,font=\sansmath\sffamily]
    \tkzDefPoint(0,1){A}
    \tkzDefPoint(4,0){B}
    \tkzDefPoint(3,6){C}

    \tkzLabelPoints[below left,font=\sansmath\sffamily](A)
    \tkzLabelPoints[below right,font=\sansmath\sffamily](B)
    \tkzLabelPoints[above,font=\sansmath\sffamily](C)

    % Draw polygon
    \tkzDrawPolygon[thick,fill=gray!10](A,B,C)

    % Get mid points
    \tkzDefMidPoint(A,B) \tkzGetPoint{F}
    \tkzDefMidPoint(A,C) \tkzGetPoint{E}
    \tkzDefMidPoint(B,C) \tkzGetPoint{D}

    \tkzDefCircle[circum,/tikz/overlay](D,E,F)\tkzGetPoint{Icircle}\tkzGetLength{rIN}

    % Get 'Hoehe'
    \tkzInterLC[/tikz/overlay](B,C)(A,B)\tkzGetPoints{hah1}{hah2}
    \tkzDefMidPoint(hah1,hah2) \tkzGetPoint{G}

    \tkzInterLC[/tikz/overlay](A,B)(C,A)\tkzGetPoints{hch1}{hch2}
    \tkzDefMidPoint(hch1,hch2) \tkzGetPoint{H}

    \tkzInterLC[/tikz/overlay](A,C)(B,A)\tkzGetPoints{hbh1}{hbh2}
    \tkzDefMidPoint(hbh1,hbh2) \tkzGetPoint{I}

    % 'Hoehenschnittpunkt'
    \tkzInterLL(A,G)(C,H)\tkzGetPoint{S}

    % More points
    \tkzInterLC[/tikz/overlay,R](A,G)(Icircle,\rIN pt)\tkzGetPoints{G}{J}
    \tkzInterLC[/tikz/overlay,R](B,I)(Icircle,\rIN pt)\tkzGetPoints{L}{I}
    \tkzInterLC[/tikz/overlay,R](C,H)(Icircle,\rIN pt)\tkzGetPoints{H}{K}

    % Draw polygon
    \tkzDrawPolygon[thick](A,B,C)
    \tkzDefCircle[in,/tikz/overlay](A,B,C) \tkzGetPoint{Ocircle}\tkzGetLength{dOcircle}
    \tkzDrawCircle[R](Ocircle,\dOcircle pt)

    \tkzDefCircle[circum,/tikz/overlay](D,E,F)\tkzGetPoint{Icircle}\tkzGetLength{rIN}
    \tkzClipCircle[R](Icircle,5cm)

    % Get escribed circles
    % first one
    \tkzDefLine[bisector](B,A,C)\tkzGetPoint{alphaBisector}
    \tkzDefBarycentricPoint(A=1,B=-2)\tkzGetPoint{ABproj}
    \tkzDefLine[bisector](ABproj,B,C)\tkzGetPoint{betaOuterBisector}
    % \tkzDrawSegments[blue](A,alphaBisector B,betaOuterBisector)
    \tkzInterLL(A,alphaBisector)(B,betaOuterBisector) \tkzGetPoint{EscribedCircleCenter1}
    \tkzInterLC(A,alphaBisector)(Icircle,H)\tkzGetPoints{EscribedCircleRadius1a}{EscribedCircleRadius2b}
    \tkzDrawCircle[color=blue](EscribedCircleCenter1,EscribedCircleRadius1a)

    % second one
    \tkzDefLine[bisector](C,B,A)\tkzGetPoint{betaBisector}
    \tkzDefBarycentricPoint(B=1,A=-2)\tkzGetPoint{BAproj}
    \tkzDefLine[bisector](C,A,BAproj)\tkzGetPoint{alphaOuterBisector}
    % \tkzDrawSegments[blue](B,betaBisector A,alphaOuterBisector)
    \tkzInterLL(B,betaBisector)(A,alphaOuterBisector) \tkzGetPoint{EscribedCircleCenter2}
    \tkzInterLC(B,betaBisector)(Icircle,H)\tkzGetPoints{EscribedCircleRadius2a}{EscribedCircleRadius2b}
    \tkzDrawCircle[color=blue](EscribedCircleCenter2,EscribedCircleRadius2b)

    % third one
    \tkzDefLine[bisector](A,C,B)\tkzGetPoint{gammaBisector}
    \tkzDefBarycentricPoint(A=-2,C=1)\tkzGetPoint{ACproj}
    \tkzDefLine[bisector](ACproj,A,B)\tkzGetPoint{gammaOuterBisector}
    % \tkzDrawSegments[blue](C,gammaBisector A,gammaOuterBisector)
    \tkzInterLL(C,gammaBisector)(A,gammaOuterBisector) \tkzGetPoint{EscribedCircleCenter3}
    \tkzInterLC(C,gammaBisector)(Icircle,H)\tkzGetPoints{EscribedCircleRadius3a}{EscribedCircleRadius3b}
    \tkzDrawCircle[color=blue](EscribedCircleCenter3,EscribedCircleRadius3b)


    \tkzDrawPoints[size=2,fill=black](D,E,F,Ocircle)
    \tkzDrawPoints[size=2,fill=red,color=red](Icircle)
    \tkzDrawCircle[R,color=red,very thick](Icircle,\rIN pt)
\end{tikzpicture}
\end{preview}
\end{document}