Let $A_1A_2...A_{2018}$ be regular $2018$-gon. Radius of it's circumcircle is $R$. Prove that: $$A_1A_{1008}-A_1A_{1006}+A_1A_{1004}-A_1A_{1002} + ... + A_1A_4 -A_1A_2=R$$
Source: Latvian TST for Baltic Way 2019, Problem 11
Tags: geometry, circumcircle
Let $A_1A_2...A_{2018}$ be regular $2018$-gon. Radius of it's circumcircle is $R$. Prove that: $$A_1A_{1008}-A_1A_{1006}+A_1A_{1004}-A_1A_{1002} + ... + A_1A_4 -A_1A_2=R$$