Titlebook: Geometric Transformations; R?zvan Gelca,Ionu? Oni?or,Carlos Yuzo Shine Textbook 2022 The Editor(s) (if applicable) and The Author(s), unde

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https://doi.org/10.1007/978-3-662-00516-3. In all solutions the triangle . is oriented counterclockwise. The first solution uses . and can be followed on Fig. 12.1. Let .., .., and .. be the centers of the equilateral triangles ..., ..., and ..., respectively.

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IsometriesOur discussion begins with those transformations that lie at the heart of the notion of equality in geometry. We tend to identify two geometric figures and call them . (or . when we are very rigorous) if we can place one on top of the other to coincide exactly.

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IsometriesWhat is the composition of two reflections? Where does it map the first triangle?

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IsometriesThe composition of the two reflections is either a translation or a rotation, and this transformation maps the triangle . to itself. It cannot be a translation because its repeated applications would move the triangle away from itself. Therefore, it is a rotation. The triangle that is invariant under some rotation is the equilateral triangle.

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InversionsPlacing the center of inversion at the origin of the coordinate system, the inversions have the equations . and ., so their composition has the equation ., which is the homothety of center . and radius ..

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