\documentclass{standalone}
\usepackage{luamplib}
\begin{document}
\mplibtextextlabel{enable}
\begin{mplibcode}
input geometry-tangent-times
vardef adjust_time(expr tt, AA, BB) =
  tt + 1/45 angle (point 0 of AA - center AA)
  - 1/45 angle (point 0 of BB - center BB)
enddef;
beginfig(1);
path A, B; % randomly rotated to show off "adjust_time"
A = fullcircle scaled 144 rotated uniformdeviate 360;
B = fullcircle scaled 60 shifted 240 right rotated 36;

numeric R, r;
R = abs (point 0 of A - center A);
r = abs (point 0 of B - center B);

path C;
C = fullcircle scaled (2R+2r) shifted center A; % NB +ve

numeric t, t', t'', u, u', u'';
(t, u) = tangent_times(C, center B);
t' = adjust_time(t, C, A);
u' = adjust_time(u, C, A);
t'' = adjust_time(t + 4, C, B);  % Note the plus fours
u'' = adjust_time(u + 4, C, B);

draw A withpen pencircle scaled 2 withcolor 3/4[blue, white];
draw B withpen pencircle scaled 2 withcolor 3/4[blue, white];
draw C withpen pencircle scaled 1 withcolor 3/4[blue, white];

draw subpath (t', u') of A -- subpath (u'', t'') of B -- cycle;
draw center B -- subpath (t, u) of C -- cycle dashed evenly;

draw center B -- point t'' of B dashed withdots scaled 1/2;
draw center B -- point u'' of B dashed withdots scaled 1/2;
draw point t of C -- point t' of A dashed withdots scaled 1/2;
draw point u of C -- point u' of A dashed withdots scaled 1/2;

dotlabel.ulft(btex $t$   etex, point t   of C);
dotlabel.lrt (btex $t'$  etex, point t'  of A);
dotlabel.lrt (btex $t''$ etex, point t'' of B);
dotlabel.lrt (btex $u$   etex, point u   of C);
dotlabel.ulft(btex $u'$  etex, point u'  of A);
dotlabel.ulft(btex $u''$ etex, point u'' of B);
drawdot center B withpen pencircle scaled dotlabeldiam;

drawoptions(withcolor 1/2[blue, white]);
  label.urt(btex $A$ etex, point 1/2(t'+u'- 7.6) of A);
  label.rt (btex $B$ etex, point 1/2(t''+u''- 2) of B);
  label.urt(btex $C$ etex, point 1/2(t+u-8) of C);
drawoptions();
endfig;
\end{mplibcode}
\end{document}