# The Broken Chord Theorem

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A Mathematical Droodle

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Copyright © 1996-2018 Alexander BogomolnyThe applet purports to remind of 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:

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? |

Extend AC to F so that CF = BC. ΔBCF isosceles. Let α be its base angle:

(2) | ∠APB = 2α = 2∠CFB = 2∠AFB. |

In the circumcircle of ΔABF, P, being the midpoint of the arc ACB, lies on the perpendicular to chord AB and also ∠APB = 2∠AFB. Which implies that APB is the central angle subtending chord AB. P therefore is the center of that circle in which AF is a chord. The perpendicular from P to AF (read AC) crosses AF at its midpoint and (1) follows.

### References

- R. Honsberger,
*Episodes in Nineteenth and Twentieth Century Euclidean Geometry*, MAA, 1995, pp. 1-2

### 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 Bogomolny71999810