\input{preamble-standalone.ltx}
\begin{document}

% Ex. No. 52 (Section  9.2.2 : Example of \tkzname{homothety} and \tkzname{projection})

\begin{tikzpicture}[scale=1.2]
  \tkzDefPoint(0,1){A}   \tkzDefPoint(5,3){B}   \tkzDefPoint(3,4){C}
  \tkzDefLine[bisector](B,A,C)                     \tkzGetPoint{a}
  \tkzDrawLine[add=0 and 0,color=magenta!50 ](A,a)
  \tkzDefPointBy[homothety=center A ratio .5](a)   \tkzGetPoint{a'}
  \tkzDefPointBy[projection = onto A--B](a')       \tkzGetPoint{k'}
  \tkzDefPointBy[projection = onto A--B](a)       \tkzGetPoint{k}
  \tkzDrawLines[add= 0 and .3](A,k A,C)
  \tkzDrawSegments[blue](a',k' a,k)
  \tkzDrawPoints(a,a',k,k',A)
  \tkzDrawCircles(a',k' a,k)
  \tkzLabelPoints(a,a',k,A)
\end{tikzpicture}

\end{document}