# The Broken Chord Theorem

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A Mathematical Droodle

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Copyright © 1996-2018 Alexander BogomolnyThe 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. |

What if applet does not run? |

Join P to B and observe that

FM = MC.

But, by construction, also

AF = BC.

Adding the two gives the desired result

### References

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

### The Broken Chord Theorem

- The Broken Chord Theorem: Proof Close to Archimedes'
- The Broken Chord Theorem: proof by Gregg Patruno
- The Broken Chord Theorem by Paper Folding
- The Broken Chord Theorem: proof by Stuart Anderson
- The Broken Chord Theorem: proof by Bui Quang Tuan
- The Broken Chord Theorem: proof by Mariano Perez de la Cruz
- Pythagoras' from the Star of David
- Pythagoras' from Broken Chords
- Extremal Problem in a Circular Segment

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Copyright © 1996-2018 Alexander Bogomolny71672744