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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. |
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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Proof by Stuart Anderson
Extend PM to meet the circle at Q. Assume Q is different from B and let BQ and AC extended intersect in F. In DAQF, QM is the altitude from vertex Q. It is also the angle bisector at Q because P is the midpoint of chord APB, so that the angles at Q subtend equal arcs:
 AQP = FQP (= BQP).
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We see that DAQF is isosceles and
Now, since quadrilateral AQBC is cyclic its opposite angles add to p. On the other hand, its angles at B and C are supplementary to those in DAQF. It follows that DBCF is similar to DAQF and, hence, is also isosceles:
(At the outset we assumed that Q is different from B. If they coincide then instead of BQ we consider the tangent to the circle at B.)
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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