  We revisit a standard polygon containment problem: given a convex $k$-gon $P$
and a convex $n$-gon $Q$ in the plane, find a placement of $P$ inside $Q$ under
translation and rotation (if it exists), or more generally, find the largest
copy of $P$ inside $Q$ under translation, rotation, and scaling.
  Previous algorithms by Chazelle (1983), Sharir and Toledo (1994), and
Agarwal, Amenta, and Sharir (1998) all required $\Omega(n^2)$ time, even in the
simplest $k=3$ case. We present a significantly faster new algorithm for $k=3$
achieving $O(n$polylog $n)$ running time. Moreover, we extend the result for
general $k$, achieving $O(k^{O(1/\varepsilon)}n^{1+\varepsilon})$ running time
for any $\varepsilon>0$.
  Along the way, we also prove a new $O(k^{O(1)}n$polylog $n)$ bound on the
number of similar copies of $P$ inside $Q$ that have 4 vertices of $P$ in
contact with the boundary of $Q$ (assuming general position input), disproving
a conjecture by Agarwal, Amenta, and Sharir (1998).
