Isosceles on the Sides of a Triangle: What is this about?
A Mathematical Droodle


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Explanation

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Copyright © 1996-2017 Alexander Bogomolny

The applet's purpose is to suggest the following statement:

  On the sides of ΔABC erect isosceles triangles AFB, BDC, and CEA. Prove that the perpendiculars through A, B, C to EF, FD, DE, respectively are concurrent.


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.


What if applet does not run?

Proof

The proof is straightforward. Consider three circles cD, cE, cF centered at D, E, F with radii DB = DC, EC = EA, and FA = FB, respectively. cD and cE meet in C and some other point, say, c so that Cc is their common chord and serves therefore their radical axis. The chord Cc is orthogonal to the line of centers, DE. That is, Cc coincides with the perpendicular from C to DE. Similarly, the other two perpendiculars, serve as the radical axes of the other two pairs of the circles. But the radical axes of three circles concur in a point, the radical center of the circles. So do the perpendiculars at hand.

References

  1. R. Honsberger, Mathematical Delights, MAA, 2004, pp. 19-20

Radical Axis and Radical Center

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Copyright © 1996-2017 Alexander Bogomolny

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