Mark the tangency point on each of the three tangent lines

From either $P$ or $Q,$ tangent segments to the same circle have equal lengths

Let the internal tangent touch $O$ and $O'$ at $R$ and $S.$ Let the external tangent through $P$ touch them at $X$ and $Y,$ and let the one through $Q$ touch them at $V$ and $W.$ Equal tangent segments from a point give $$\begin{gathered} PR=PX,\\ PS=PY,\\ QR=QV,\\ QS=QW. \end{gathered}$$ Along the internal tangent, $PR+QR=PQ$ and $PS+QS=PQ.$ Adding, $$\begin{aligned} PR+QR+PS+QS=2PQ. \end{aligned}$$ Along the external tangents, $PX+PY=XY$ and $QV+QW=VW.$ Therefore $2PQ=XY+VW.$ The two external common tangent segments have equal length, so $XY=VW$ and hence $PQ=XY=VW.$

Therefore, the correct answer is C.