Problem
If points $P, \ Q, \ R$ and $S$ are taken on the sides $AB, \ BC, \ CD$ and $DA$ of a parallelogram $ABCD$ such that $AP=BQ=CR=DS$, then the quadrilateral $PQRS$ is also a parallelogram.
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Solution
From the diagram,
$$∠A=∠C, \qquad PA=RC \qquad and \qquad AS=CQ,$$
$$∴ \quad △APS≡△CRQ,$$
$$∴ \quad SP=QR. \qquad [1]$$
Similarly,
$$PQ=RS. \qquad [2]$$
From $[1]$ and $[2]$, the quadrilateral $PQRS$ is a parallelogram.
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Reference Teiichiro Sasabe (1976) The Encyclopedia of Geometry (2nd edition), Seikyo-Shinsha, pp.45-46.