Dissection of Triangle into Rhombus
by Hubert Shutrick
The applet below illustrates Problem 8 from the 2010 All-Russian Olympiad:
In an acute triangle ABC, the median AM is longer than side AB. Prove that you can cut the triangle into three parts out of which you can construct a rhombus.
Let M be the midpoint of BC and Bm the midpoint AC. Assume that BM < AB. Then there is point N on AB such that MN = AB. Cut over NM and BmM and swivel the two pieces 180°, one around Bm, the other around M. N' and N'' are the corresponding images of N and M' is the image of M under the rotation around Bm.
The proof that the construction leads to a rhombus is rather straightforward: all four sides of the resulting quadrilateral N'N''MM' equal AB.
(There is another attempt at solving the problem.)
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