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Directly Similar Figures: What is that about?
A Mathematical Droodle


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What if applet does not run?

Explanation

Copyright © 1996-2009 Alexander Bogomolny

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Explanation

The applet suggests the following statment:

(1) Given two similar triangles ABC and A'B'C' and a point P, let A1 = P + AA', B1 = P + BB', and C1 = P + CC'. Then triangle A1B1C1 is similar to ABC and A'B'C'.


This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


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What if applet does not run?

The statement implies something more general, akin to the Fundamental Theorem of Directly Similar Figures and relates to the properties of spiral similarities. For any two directly similar figures S and S', if A is a generic point of S and A' is the corresponding point of S', then for any point P, P + AA' traces a figure similar two both S and S'.

The proof (in complex numbers) follows easily from

 

with m = 1 and l = -1.

References

  1. I. M. Yaglom, Geometric Transformations II, MAA, 1968

Copyright © 1996-2009 Alexander Bogomolny

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