# The Broken Chord TheoremWhat is this about? A Mathematical Droodle

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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:

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.

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

 (1) AM = MC + BC.

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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:

FM = MC.

But, by construction, also

AF = BC.

Adding the two gives the desired result

### References

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