Points on a unit circle

Let S be a unit circle, that is, the circle of radius 1 centered at the origin in Euclid space R^2. Let a1, a2, …, an be points on S. Prove that there exists a point p on S such that |a1-p| × |a2 - p| × … × |an - p| = 1. (Here, |a1-p| represents the distance between a1 and p.)

Comments

This question is an exercise in this book: Complex Analysis,

Comments 2

Shuho Kanda (the university of Tokyo) told me that there exists a point p on S such that |a1-p| × |a2 - p| × … × |an - p| = 2.