\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(0,0){A}
    \tkzDefPoint(6,0){B}
    \tkzDefPoint(1,4){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)

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

    \tkzInterLC[R,/tikz/overlay](B,C)(Ocircle,\dOcircle pt)\tkzGetPoints{X}{X2}
    \tkzInterLC[R,/tikz/overlay](A,C)(Ocircle,\dOcircle pt)\tkzGetPoints{Y}{Y2}
    \tkzInterLC[R,/tikz/overlay](A,B)(Ocircle,\dOcircle pt)\tkzGetPoints{Z}{Z2}
    \tkzInterLL(A,X)(B,Y)\tkzGetPoint{G}

    \tkzDrawSegments[dashed](A,X B,Y C,Z)

    \tkzDrawPoints[size=2,fill=black](Ocircle,X,Y,Z)
    \tkzDrawPoints[size=2,fill=red,color=red](G)

    \tkzLabelPoints[above right,font=\sansmath\sffamily](X)
    \tkzLabelPoints[left,font=\sansmath\sffamily](Y)
    \tkzLabelPoints[below,font=\sansmath\sffamily](Z)
    \tkzLabelPoints[red,above left,font=\sansmath\sffamily](G)
\end{tikzpicture}
\end{preview}
\end{document}