The centers of the circles are at intersections of auxiliary circles around the other circles where the radii of the auxiliary circles are given by the original radii plus the radius of the new circle.
\documentclass[tikz,border=3mm]{standalone}
\usetikzlibrary{intersections}
\begin{document}
\begin{tikzpicture}[dot/.style={circle,fill,inner sep=1pt},
declare function={R=5;rA=1.5;rB=1;rC=1.3;}]
\draw (0,0) coordinate(O) circle[radius=R];
\draw (160:R-rA) node[dot,label=20:$A$](A){} circle[radius=rA];
\path[overlay,name path=auxB1] (O) circle[radius=R-rB];
\path[overlay,name path=auxB2] (A) circle[radius=rA+rB];
\draw[name intersections={of=auxB1 and auxB2,by={aux,B}}]
(B) node[dot,label=above:$B$]{} circle[radius=rB];
\path[overlay,name path=auxC1] (A) circle[radius=rA+rC];
\path[overlay,name path=auxC2] (B) circle[radius=rB+rC];
\draw[name intersections={of=auxC1 and auxC2,by={aux,C}}]
(C) node[dot,label=above:$C$]{} circle[radius=rC];
\end{tikzpicture}
\end{document}

Of course, no library is needed for this, only the cosine law which tells us the angles of a triangle with given edge lengths.
\documentclass[tikz,border=3mm]{standalone}
\begin{document}
\begin{tikzpicture}[dot/.style={circle,fill,inner sep=1pt},
declare function={R=5;rA=1.5;rB=1;rC=1.3;alpha=160;
cosinelaw(\a,\b,\c)=acos((\a*\a+\b*\b-\c*\c)/(2*\a*\b));}]
\draw (0,0) coordinate(O) circle[radius=R];
\draw (alpha:R-rA) node[dot,label=alpha-180:$A$](A){} circle[radius=rA];
\draw ({-cosinelaw(R-rA,R-rB,rA+rB)+alpha}:R-rB)
node[dot,label=above:$B$](B){} circle[radius=rB];
\pgfmathsetmacro{\myturn}{cosinelaw(rA+rB,rC+rB,rC+rA)-180}
\path (A) -- (B) -- ([turn]\myturn:rB+rC)
node[dot,label=above:$C$](C){};
\draw (C) circle[radius=rC];
\end{tikzpicture}
\end{document}
