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Rotations in Disguise: What Is This About?
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Copyright © 1996-2008 Alexander Bogomolny
Rotations in Disguise
In one of O. Bottema's theorem, squares have been constructed on the sides of a triangle and a line joining two corners of the squares passed through the center of the square on the third side with the opposite orientation.
The applet below suggests an analogue of Bottema's theorem wherein equilateral triangles are constructed on the sides AC, BC of ΔABC outside the latter and another one on the side AB, with opposite orientation. The triangles are denoted AB'C, A'BC, and ABC'. If M'is the center of ΔABC', then
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The assertion is an immediate consequence of a property of rotations. The product of two rotations is a rotation through the angle equal to the sum of the two rotation angles and the center can be easily obtained at the intersection of two rotation axes.
To apply this reasoning, consider two rotation through 60o: one centered at A, the other at B. B' first rotated around A into C which then continues around B into A'. The product is a rotation through 120o with the center at the center of ΔABC'.
Copyright © 1996-2008 Alexander Bogomolny
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