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The Broken Chord Theorem: What is this about?
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The 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. |
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Join P to B and observe that 
PAC = PBC
| 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
- A Proof Close to Archimedes'
- A Proof by Gregg Patruno
- A Proof by Paper Folding
- A Proof by Stuart Anderson
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