The Broken Chord Theorem
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A Mathematical Droodle
Explanation
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Copyright © 1996-2012 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
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:
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
- 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
|Activities|
|Contact|
|Front page|
|Contents|
|Geometry|
|Store|
Copyright © 1996-2012 Alexander Bogomolny
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