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 Bogomolny71471595