Chords of a Unit Circle

Take N equidistant points on the unit circle. Pick one of those points, then draw chords from it to all other points on the circle, as in Figure 1. What is the product of lengths of those chords?Surprise: for N points, the product is just N. Pretty cool, huh?

Presentation Suggestions:
Do simple cases first and draw pictures: for N=2, there is just one chord (the diameter of the circle, length 2). For N=3, there are two chords (both of length Sqrt[3], product=3). For N=4, there are 3 chords (one diameter of length 2, two of length Sqrt[2], product=4). Have people conjecture the answer for themselves!

The Math Behind the Fact:
If you know about complex numbers, you can prove this by letting the points of the circle be given by roots of unity, and letting z be any point (not necessarily the unit disc) and looking at the product of lengths of the chords from z to all N points on the circle. Divide this expression by (z-1), and take the limit as z goes to 1, and voila!, you will get N.

If you liked this Fun Fact, now see what happens with Chords of an Ellipse.

How to Cite this Page: 
Su, Francis E., et al. “Chords of a Unit Circle.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

Fun Fact suggested by:
Francis Su

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