The Broken Chord Theorem
Proof by Mariano Perez de la Cruz

Proof

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By extending PM to its intersection with the circumference we get point S. A line from S through B intersects the AC extended in F.

∠MSA = ∠MSF since P bisects the original arc AB, implying AM = MF.

We are to prove that CF = CB.

This is so because, ∠CBF is suplementary of ∠CBS, and, since ACBS is an inscribed quadrilateral, ∠CAS = ∠CFB, both being suplementary to ∠CBS. Therefore CF = CB.

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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