# The Broken Chord Theorem

by Paper Folding

The applet below is an illustration of an elegant theorem credited to Archimedes.

Assume for the definiteness' sake that AC > BC, so that M lies on AC and the theorem states that

(1) | AM = MC + BC. |

To get an insight into the likely origins of the problem, we'll start with an isosceles triangle APF in which

This shows that the configuration in the Broken Chord theorem is naturally arrived at by paper folding. For a proof one only needs to unfold the paper. Or so it seems ...

What if applet does not run? |

We may start with extending AC to F, as in another proof. But first note that the problem presents several pairs of inscribed angles subtended by the same arcs. For this proof, we are interested in three pairs (all different from the one used elsewhere):

α = ∠PAC = ∠PBC, β = ∠ABP = ∠ACP, γ = ∠BAC = ∠BPC. |

The three angles are not independent. Since ΔAPB is isosceles we observe that

(2) | α + γ = β. |

We now proceed with the construction. Let F lie on the extension of AC so that CF = BC. ΔBCF is isosceles. Its apex angle is exterior to ΔABC, from which we conclude that

∠BCF = α + β + γ = 2β.

If T is the point (not shown in the applet) where PC crosses BF, then ∠FCT is vertical to

Nathan Bowler came up with a shorter and a more direct argument. First observe that

∠BAP + ∠BCP = π.

But

∠BAP = ∠ABP = β = ∠ACP.

So it follows that PC is the external angle bisector at C of ΔBCF. Which, in turn, implies that the reflection F of B falls on the extension of AC. A shorter way to unfolding!

As an extra insight, observe that we might have reflected in PC vertex A instead of B. This would produce another isosceles triangle BPG equal to APF. A proof may now be built based the fact that the triangles are obtained from each other by a rotation around P.

### References

- S. E. Louridas, M. Th. Rassias,
*Problem-Solving and Selected Topics in Euclidean Geometry*, Springer, 2013 (69-71)

### 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

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- Angle Trisection by Paper Folding
- Angles in Triangle Add to 180
^{o} - Broken Chord Theorem by Paper Folding
- Dividing a Segment into Equal Parts by Paper Folding
- Egyptian Triangle By Paper Folding
- Egyptian Triangle By Paper Folding II
- Egyptian Triangle By Paper Folding III
- My Logo
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- Parabola by Paper Folding
- Radius of a Circle by Paper Folding
- Regular Pentagon Inscribed in Circle by Paper Folding
- Trigonometry by Paper Folding
- Folding Square in a Line through the Center
- Tangent of 22.5
^{o}- Proof Without Words - Regular Octagon by Paper Folding
- The Shortest Crease
- Fold Square into Equilateral Triangle
- Circle Center by Paperfolding
- Folding and Cutting a Square

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