A friend of me Torbjørn, showed me how to build the image below using tikz

with the following code.
\documentclass{article}
\usepackage{tkz-euclide}
\usetkzobj{all}
\begin{document}\begin{minipage}{0.5\textwidth}
I en $\Delta ABC$ er $\angle A = 90^{\circ}$ En sirkel med sentrum i
$S$ er innskrevet i trekanten. Sidene $AC$ og $BC$ tangerer sirkelen
i punktene $D$ og $E$. Linjen gjennom $B$ og $S$ skjærer $DE$ i $F$. \\
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Se skissen til høyre. \\
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Du får oppgitt at $DC=EC$.\\
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Vi setter $\angle ABC = v, \: \angle BCU = u, \: $ og $\angle BFE=x$\\
\end{minipage}
\begin{minipage}{0.5\textwidth}
\begin{tikzpicture}[scale=1.5]
\tkzInit[ymin=-1,ymax=5,xmin=-1,xmax=5]
\tkzClip
\tkzDefPoint(0,0){A}
\tkzDefPoint(4,0){B}
\tkzDefPoint(0,4){C}
\tkzDefCircle[in](A,B,C)
\tkzGetPoint{S}
\tkzGetLength{rSN}
\tkzDrawPoints(A,B,C)
\tkzDrawCircle[R](S,\rSN pt)
\tkzDrawPolygon[very thick](A,B,C)
\tkzInterLC[R](A,C)(S,\rSN pt) \tkzGetFirstPoint{D}
\tkzInterLC[R](B,C)(S,\rSN pt) \tkzGetFirstPoint{E}
\tkzInterLL(B,S)(D,E) \tkzGetPoint{F}
\tkzDrawPoints(D,E,F,S)
\tkzDrawSegments[thick,red](D,E)
\tkzDrawSegments[thick,violet](B,F)
\tkzLabelPoints[left](D)
\tkzLabelPoints[above left](C,F)
\tkzLabelPoints[below right](B)
\tkzLabelPoints[below left](A,S)
\tkzLabelPoints[above right](E)
\tkzMarkAngle[size=0.4 cm](B,F,E)
\tkzLabelAngle[pos=0.6](B,F,E){\large $x$}
\tkzMarkRightAngle(B,A,C)
\end{tikzpicture}
\end{minipage}
\end{document}
Using geogebra, I was able to prduce a simmilar looking image, as below

My question is. How can I transform my tikz image into looking more like the geogebra image? The point in S should be black, not violet, and one should keep the angle A. (I had problems both changing the color of the inscribed circle, changing the size of the labels, and changing the color of the points.)
I think that was it for now. Alas the tikz image works better without the minipages.

