\begin{tikzpicture}[scale=2]
\clip (-1.1,-1.1) rectangle (1.7,1.1);
\coordinate (o) at (0,0);
\coordinate (a) at (1,0);
\def\r{.65};
\tkzDefPoint(0:(1+\r^2)/(2*\r)){b}
\tkzDrawCircle[dashed,color=blue](o,a)
\foreach \x in {0,1,...,7} \tkzDefPoint(\x*45:\r){\x};
\tkzDefPoint(0:1/\r){vo*}
%\coordinate (vo) at (0:.8409);
\hypsegment{1}{0}{o}{a}
\hypsegment{2}{1}{o}{a}
\hypsegment{3}{2}{o}{a}
\hypsegment{4}{3}{o}{a}
\hypsegment{5}{4}{o}{a}
\hypsegment{6}{5}{o}{a}
\hypsegment{7}{6}{o}{a}
\hypsegment{0}{7}{o}{a}
\draw (o)--(vo*);
\draw[thin](1)--(0);
\draw[dashed,->](o)--(22.5:1.7);
\tkzDefLine[perpendicular=through b,K=.7](o,b)\tkzGetPoint{c}
\draw[dashed](b)--(c);
\tkzDefMidPoint(0,1)\tkzGetPoint{m}
\tkzInterLL(b,c)(o,m)\tkzGetPoint{zo}
\tkzDrawPoints[fill=red](o,0,1,2,3,4,5,6,7)
\tkzDrawPoints(b,vo*,m,zo)
\tkzLabelPoint[below right= 0pt and -2pt](b){$$b$$}
\tkzLabelPoint[below](vo*){$$v_o^*$$}
\tkzLabelPoint[below right=-2pt](0){$$v_0$$}
\tkzLabelPoint[above](1){$$v_1$$}
\tkzLabelPoint[above](2){$$v_2$$}
\tkzLabelPoint[left](3){$$v_3$$}
\tkzLabelPoint[left](4){$$v_4$$}
\tkzLabelPoint[below](5){$$v_5$$}
\tkzLabelPoint[below](6){$$v_6$$}
\tkzLabelPoint[below](7){$$v_7$$}
\tkzLabelPoint[right](zo){$$z_0$$}
\end{tikzpicture}