O is the center of a regular n-gon P_{1}P_{2} ... P_{n} and X is a point outside the n-gon on the line OP_{1}. Show that XP_{1} XP_{2} ... XP_{n} + OP_{1}^{n} = OX^{n}.

**Solution**

*Easy.*

Evidently it is sufficient to prove the result for the case OP_{1} = 1. So represent P_{k} by the complex number ω^{k}, where ω = e^{i 2π/n} an nth root of unity. Represent X by the real number r. Then we have to show that |r - 1||r - ω| ... |r - ω^{n-1}| = r^{n} - 1, but that follows immediately because r^{n} - 1 = (r - 1)(r - ω) ... (r - ω^{n-1}).

© John Scholes

jscholes@kalva.demon.co.uk

26 Nov 1999