Intouch Triangle in Poncelet Porism: What is this about?
A Mathematical Droodle

 

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Explanation

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Copyright © 1996-2018 Alexander Bogomolny

The applet attempts to illustrate a property of the intouch triangles involved in the Poncelet porism: let ABC be a generic triangle inscribed in and circumscribed around two given circles, O(R) and I(r). If d, the distance between the centers of the circles satisfies 1/(R - d) + 1/(R + d) = 1/r, there is a continuum of such triangles. Construct, for each such triangle, the corresponding intouch triangle KLM. And, for the latter, find the orthocenter H and the (bary)center G. Surprisingly, for the given two circles, neither depends on the selection of triangle ABC. Since the incenter of ΔABC which serves as the circumcenter of ΔKLM is also fixed, we may conclude that the entire Euler line is fixed point-by-point implying, for example, that the 9-point center of ΔKLM is also independent of the position of ΔABC.

 

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 https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

... to be continued ...

 

References

  1. L. Emelyanov and T. Emelyanova, Eulerís Formula and Ponceletís Porism, Forum Geometricorum, Volume 1 (2001) 137Ė140
  2. W. Gallatly, The Modern Geometry of the Triangle, Scholarly Publishing Office, University of Michigan Library (December 20, 2005)
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|Activities| |Contact| |Front page| |Contents| |Geometry|

Copyright © 1996-2018 Alexander Bogomolny
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