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The Broken Chord Theorem: What is this about?
A Mathematical Droodle


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Explanation

Copyright © 1996-2009 Alexander Bogomolny

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The applet purports to remind a theorem going under the name of The Broken Chord Theorem. The theorem is credited to Archimedes himself, although it does not appear in his Book of Lemmas:

  On the circumcircle of triangle ABC, point P is the midpoint of the arc ACB. PM is perpendicular to the longest of AC or BC. Prove that M divides the broken line ABC in half.

The proof below belongs to Gregg Patruno.

Assume for the definiteness' sake that AC > BC, so that M lies on AC and the theorem states that

(1) AM = MC + BC.


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?

Join P to B and observe that PAC = PBC since both are subtended by the same chord PC. Also, since P is the midpoint of the arc ACB, AP = BP. Find F on AC such that AF = BC. Triangles BPC and APF are equal by SAS. Their third sides are therefore also equal: FP = CP. Which means that triangle FPC is isosceles and PM is both the altitude and the median from the apex:

  FM = MC.

But, by construction, also

  AF = BC.

Adding the two gives the desired result

References

  1. R. Honsberger, More Mathematical Morsels, MAA, 1991, pp. 31-32

The Broken Chord Theorem

  1. A Proof Close to Archimedes'
  2. A Proof by Gregg Patruno
  3. A Proof by Paper Folding
  4. A Proof by Stuart Anderson

Copyright © 1996-2009 Alexander Bogomolny

34222692Page copy protected against web site content infringement by Copyscape


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