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The Broken Chord Theorem: What is this about?
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The 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:
| 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. |
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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Extend AC to F so that CF = BC. DBCF isosceles. Let a be its base angle: 
CFB = BFC.
| (2) |
APB = 2a = 2CFB = 2AFB.
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In the circumcircle of DABF, P, being the midpoint of the arc ACB, lies on the perpendicular to chord AB and also APB = 2AFB. 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
- A Proof Close to Archimedes'
- A Proof by Gregg Patruno
- A Proof by Paper Folding
- A Proof by Stuart Anderson
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