The Broken Chord Theorem
What is this about?
A Mathematical Droodle


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Explanation

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Copyright © 1996-2017 Alexander Bogomolny

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 ΔAQF, 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).

We see that ΔAQF is isosceles and AM = MF.

Now, since quadrilateral AQBC is cyclic its opposite angles add to π. On the other hand, its angles at B and C are supplementary to those in ΔAQF. It follows that ΔBCF is similar to ΔAQF and, hence, is also isosceles: BC = CF.

(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

  1. The Broken Chord Theorem: Proof Close to Archimedes'
  2. The Broken Chord Theorem: proof by Gregg Patruno
  3. The Broken Chord Theorem by Paper Folding
  4. The Broken Chord Theorem: proof by Stuart Anderson
  5. The Broken Chord Theorem: proof by Bui Quang Tuan
  6. The Broken Chord Theorem: proof by Mariano Perez de la Cruz
  7. Pythagoras' from the Star of David
  8. Pythagoras' from Broken Chords
  9. Extremal Problem in a Circular Segment

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Copyright © 1996-2017 Alexander Bogomolny

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