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

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

|Activities| |Contact| |Front page| |Contents| |Geometry|
Copyright © 1996-2018 Alexander Bogomolny72258653