How can I label points in tkz-euclide without serifes?

Question

I have the following image

As you can see, the points $A, B, C$ have serifes. I would like them to be without serifes. I do I change the font for \tkzLabelPoints?. (font=\sffamily\sansmath does not work)

MWE

\documentclass[varwidth=true, border=2pt]{standalone}

\usepackage{tkz-euclide}
\usetkzobj{all}
% Thanks to Jan Hlavacek for giving an example for hyperbolic triangles
% (http://tex.stackexchange.com/a/16749/5645)

% Thanks to Jonas Granholm for help with the angles
% (http://tex.stackexchange.com/a/204435/5645)

\usepackage{helvet}
\usepackage[eulergreek]{sansmath}

\begin{document}
\begin{tikzpicture}[scale=3,font=\sffamily\sansmath]
    \tkzDefPoint(0,0){O}
    \tkzDefPoint(1,0){Z}

    % Define points
    \tkzDefPoint(-0.7,-0.1){A}
    \tkzDefPoint(0.4,-0.3){B}
    \tkzDefPoint(0.1,0.4){C}

    % Draw big circle
    \tkzDrawCircle[fill=white](O,Z)

    % I have no idea what this does,
    % but when I remove it the rest of the image (the big circles)
    % get visible. So it seems to crop the image to the size of the circle
    \tkzClipCircle(O,Z)

    % Draw angles
    \tkzMarkAngles[fill= orange,size=0.3cm,opacity=.3](B,A,C)
    \tkzLabelAngle[pos=0.2](B,A,C){$\alpha$}

    \tkzMarkAngles[fill= orange,size=0.3cm,opacity=.3](C,B,A)
    \tkzLabelAngle[pos=0.2](C,B,A){$\beta$}

    \tkzMarkAngles[fill= orange,size=0.3cm,opacity=.3](A,C,B)
    \tkzLabelAngle[pos=0.2](A,C,B){$\gamma$}


    % "Clip" angles
    \tkzDrawCircle[fill,orthogonal through=A and B,color=white](O,Z)
    \tkzDrawCircle[fill,orthogonal through=A and C,color=white](O,Z)
    \tkzDrawCircle[fill,orthogonal through=B and C,color=white](O,Z)

    % Draw big circle
    \tkzDrawCircle(O,Z)

    % Draw lines
    \tkzDrawCircle[orthogonal through=A and B,color=green](O,Z)
    \tkzDrawCircle[orthogonal through=A and C,color=blue](O,Z)
    \tkzDrawCircle[orthogonal through=B and C,color=red](O,Z)

    % Draw points and label them
    \tkzDrawPoints[color=black,fill=red,size=5](A,B,C)
    \tkzLabelPoints[below](A)
    \tkzLabelPoints[below left](B)
    \tkzLabelPoints[above left](C)
\end{tikzpicture}
\end{document}

Answer 1

The labels of points are TikZ nodes, and you can add options to \tkzLabelPoints as with normal nodes, so doing \tkzLabelPoints[below,font=\sansmath](A) works for a single label. To set this for all point labels, you can modify the label style style, which is used for them, with

label style/.append style={font=\sansmath}

added to the tikzpicture options.

Complete code:

\documentclass[varwidth=true, border=2pt]{standalone}

\usepackage{tkz-euclide}
\usetkzobj{all}
% Thanks to Jan Hlavacek for giving an example for hyperbolic triangles
% (http://tex.stackexchange.com/a/16749/5645)

% Thanks to Jonas Granholm for help with the angles
% (http://tex.stackexchange.com/a/204435/5645)

\usepackage{helvet}
\usepackage[eulergreek]{sansmath}

\begin{document}
\begin{tikzpicture}[scale=3,label style/.append style={font=\sansmath}]
    \tkzDefPoint(0,0){O}
    \tkzDefPoint(1,0){Z}

    % Define points
    \tkzDefPoint(-0.7,-0.1){A}
    \tkzDefPoint(0.4,-0.3){B}
    \tkzDefPoint(0.1,0.4){C}

    % Draw big circle
    \tkzDrawCircle[fill=white](O,Z)

    % I have no idea what this does,
    % but when I remove it the rest of the image (the big circles)
    % get visible. So it seems to crop the image to the size of the circle
    \tkzClipCircle(O,Z)

    % Draw angles
    \tkzMarkAngles[fill= orange,size=0.3cm,opacity=.3](B,A,C)
    \tkzLabelAngle[pos=0.2](B,A,C){$\alpha$}

    \tkzMarkAngles[fill= orange,size=0.3cm,opacity=.3](C,B,A)
    \tkzLabelAngle[pos=0.2](C,B,A){$\beta$}

    \tkzMarkAngles[fill= orange,size=0.3cm,opacity=.3](A,C,B)
    \tkzLabelAngle[pos=0.2](A,C,B){$\gamma$}


    % "Clip" angles
    \tkzDrawCircle[fill,orthogonal through=A and B,color=white](O,Z)
    \tkzDrawCircle[fill,orthogonal through=A and C,color=white](O,Z)
    \tkzDrawCircle[fill,orthogonal through=B and C,color=white](O,Z)

    % Draw big circle
    \tkzDrawCircle(O,Z)

    % Draw lines
    \tkzDrawCircle[orthogonal through=A and B,color=green](O,Z)
    \tkzDrawCircle[orthogonal through=A and C,color=blue](O,Z)
    \tkzDrawCircle[orthogonal through=B and C,color=red](O,Z)

    % Draw points and label them
    \tkzDrawPoints[color=black,fill=red,size=5](A,B,C)
    \tkzLabelPoints[below](A)
    \tkzLabelPoints[below left](B)
    \tkzLabelPoints[above left](C)
\end{tikzpicture}
\end{document}