# The Broken Chord Theorem

What is this about?

A Mathematical Droodle

What if applet does not run? |

|Activities| |Contact| |Front page| |Contents| |Geometry|

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

|Activities| |Contact| |Front page| |Contents| |Geometry|

Copyright © 1996-2018 Alexander Bogomolny