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Question

Let A1,A2,....,An be vertices of an n sided regular polygon such that
1A1A2=1A1A3+1A1A4

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Solution

A1A2=modofA1A2=|A1A2|
|A1Ar|2=|z1|2[22cos2(r1)πn]
=a24sin22(r1)πnby(A)
|A1Ar|=2asin(r1)πn
1cosθ=2sin2(θ/2)
Putting r=2,3,4 we have from the given relation,
1sinπn=1sin2πn+1sin3πn
2sinπncosπnsin3πn=sinπn[sin3πn+sin2πn]
Cancelsinπnassinπn0asn1
sin4πn+sin2πn=sin3πn+sin2πn
orsin4πn=sin3πn4πn=π3πn
7πn=πorn=7

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