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We propose a new technique called Rotate-and-Kill for solving the polygoninclusion and circumscribing problems. By applying this technique, we obtain$O(n)$ time algorithms for computing (1) the maximum area triangle in a given$n$-sided convex polygon $P$, (2) the minimum area triangle enclosing $P$, (3)the minimum area triangle enclosing $P$ touching edge-to-edge, i.e. the minimumarea triangle that is the intersection of three half-planes out of the $n$half-planes defining $P$, and (4) the minimum perimeter triangle enclosing $P$touching edge-to-edge. Our algorithm for computing the maximum area triangle issimpler than its alternatives given in [Chandran and Mount, IJCGA'92] and[Kallus, arXiv'17]. Our algorithms for computing the minimum area or perimetertriangle enclosing $P$ touching edge-to-edge improve the $O(n\log n)$ or$O(n\log^2n)$ time algorithms given in [Boyce \emph{et al.}, STOC'82],[Aggarwal \emph{et al.}, Algorithmica'87], [Aggarwal and J. Park., FOCS'88],[Aggarwal \emph{et al.}, DCG'94], and [Schieber, SODA'95].

A technique for solving the polygon inclusion problems