Let us assume that O is the centre of the regular polygon and z0,z1,⋯,zn−1 represent the affixes of A1,A2,⋯,An, such that
z0=1,z1=α,z2=α2,⋯,zn−1=αn−1, where α=ei2π/n
(Since all nth roots of unity can be represented as verticies of regular polygon inscribed in a circle of unit radius)
Now, |A1Ar|2=|αr−1|2=|1−αr|2
=∣∣∣1−cos2rπn−isin2rπn∣∣∣2
=(1−cos2rπn)2+(sin2rπn)2
=2−2cos2rπn
∴n∑r=2|A1A2|2=n∑r=1(2−2cosrπn)
=2(n−1)−2[cos2πn+cos4πn+⋯cos2(n−1)πn]
=2(n−1)−2 Real part of (α+α2+⋯+αn−1)
=2(n−1)−2(−1)=2n
[∵1+α+α2+⋯+αn−1=0]
∴|A1A2|2+|A1A3|2+⋯|A1An|2=2n
Also, let E=|A1A2||A1A3|⋯|A1An|
=|1−α||1−α2||1−α3|⋯|1−αn−1|
=|(1−α)(1−α2)(1−α3)⋯(1−αn−1)|
Since, 1,α,α2,⋯,αn−1 are the roots of zn−1=0
⇒(z−1)(z−α)(z−α2)⋯(z−αn−1)=zn−1
⇒(z−α)(z−α2)⋯(z−αn−1)=zn−1z−1
=1+z+z2+⋯+zn−1
Substituting z=1, we have
=(1−α)(1−α2)(1−α3)⋯(1−αn−1)=n
∴|1−α||1−α2||1−α3|⋯|1−αn−1|=n
Hence, the value of ab=2nn=2