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

Explanation

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. A Proof Close to Archimedes'
2. A Proof by Gregg Patruno
3. A Proof by Paper Folding
4. A Proof by Stuart Anderson