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


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Explanation

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Copyright © 1996-2017 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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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. The Broken Chord Theorem: Proof Close to Archimedes'
  2. The Broken Chord Theorem: proof by Gregg Patruno
  3. The Broken Chord Theorem by Paper Folding
  4. The Broken Chord Theorem: proof by Stuart Anderson
  5. The Broken Chord Theorem: proof by Bui Quang Tuan
  6. The Broken Chord Theorem: proof by Mariano Perez de la Cruz
  7. Pythagoras' from the Star of David
  8. Pythagoras' from Broken Chords
  9. Extremal Problem in a Circular Segment

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Copyright © 1996-2017 Alexander Bogomolny

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