Monday, August 7, 2023

Archimedes and the Broken chord

The theorem of the "broken chord" is ascribed to Archimedes, although his original work has been lost. It was analyzed by the Arabic mathematician Al-Biruni in his Book on the Derivation of Chords in a Circle.

[UPDATE: I have made a translation and commentary of a German translation of this book. It is here. ]

Here is the general setup:

Let A and C be any two points on a circle. Let M be equidistant from both so that arc AM is equal to arc MC. Let B be another point on the circle, lying between A and M, so that AB < BC.

Drop the perpendicular from M to F on BC.

We claim that AB + BF = FC.

I will not spoil the fun by giving the proofs here. But these are eight constructions I know about.

Draw E such that AB = EC. (As an alternative approach, draw E such that BF = FE).

Draw the rectangle such that H is on the circle.
Extend BC such that DF = FC.
Draw E such that BF = FE and D such that BM = MD.
Draw E such that BF = FE and extend ME to G.
Extend BC and MF and draw DAG colinear.
At this point, I discovered a German translation of al Biruni's book (by Suter, link below). Therefore, I switched notation to match his figures. I can select the text in Preview, then Google Translate does a good job with it.

Extend BG as shown.

Extend the perpendicular DE as shown. Draw AG. Draw the diameter DK. (Hint: DK is perpendicular to AG).
Sources: Drakaki, al Biruni, Suter. There is a chapter in my geometry book on this. The chapter as a pdf is here, and the github repo for the book is here.