The Broken Chord Theorem
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A Mathematical Droodle

Explanation

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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:

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

Join P to B and observe that ∠PAC = ∠PBC since both are subtended by the same chord PC. Also, since P is the midpoint of the arc ACB, AP = BP. Find F on AC such that AF = BC. Triangles BPC and APF are equal by SAS. Their third sides are therefore also equal: FP = CP. Which means that triangle FPC is isosceles and PM is both the altitude and the median from the apex:

But, by construction, also

Adding the two gives the desired result

References

1. R. Honsberger, More Mathematical Morsels, MAA, 1991, pp. 31-32

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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