Let O be the centre and r be the radius of the circle passing through the vertices `A_(1),A_(2),.....A_(n)`
Then, `angleA_(1)OA_(2)=(2pi)/(n)`
alos, `OA_(1)=OA_(2)=r`
Again, by cos formula we know that
`cos ((2pi)/(n))=(OA_(1)^(2)+OA_(2)^(2)-A_(1)A_(2)^(2))/(2(OA_(1))(OA_(2)))`
`rArr cos ((2pi)/(n))=(r^(2)+r^(2)-A_(1)A_(2)^(2))/(2(r) (r))`
`rArr 2 r^(2) cos ((2pi)/(n))=2r^(2)-A_(1)A_(2)^(2)`
`rArr A_(1)A_(2)^(2)-2r^(2)-2r^(2)cos((2pi)/(n))`
`rArr A_(1)A_(2)^(2)-2r^(2)[1-cos((2pi)/(n))]`
`rArr A_(1)A_(2)^(2)-2r^(2)*2r^(2)sin((pi)/(n))`
`rArr A_(1)A_(2)^(2)-4r^(2)sin((pi)/(n))`
`rArr A_(1)A_(2)^(2)-2r^(2)sin((pi)/(n))`
Similarly, `rArr A_(1)A_(3)-2r^(2)sin((2pi)/(n))`
and `rArr A_(1)A_(4)-2r^(2)sin((3pi)/(n))`
Since, `(1)/(A_(1)A_(2))=(1)/(A_(1)A_(3))+(1)/(A_(1)A_(4))" "`[ give]
`rArr (1)/(2 sin r (pi//r))=(1)/(2 r sin (2pi//n))+(1)/(2r sin (3pi//n))`
`rArr (1)/( sin (pi//r))=(1)/( sin (2pi//n))+(1)/( sin (3pi//n))`
`rArr (1)/(sin(pi//n))=(sin((3pi)/(n))+sin ((2pi)/(n)))/(sin(2pi//n)sin (3pi//n))`
`rArr sin((2pi)/(n))*sin((3pi)/(n))=sin((pi)/(n))sin((3pi)/(n))+sin((pi)/(n))*sin((2pi)/(n))`
`rArr sin((2pi)/(n))[sin((3pi)/(n))-sin((pi)/(n))]=sin((pi)/(n))*sin((3pi)/(n))`
`sin ((2pi)/(n))[{ 2 cos((3pi+pi)/(2n))sin((3pi-pi)/(2n))}]=sin((pi)/(2))*sin((3pi)/(n))`
`rArr sin((2pi)/(n))cos((2pi)/(n))*sin((pi)/(n))=sin((pi)/(n))sin((3pi)/(n))`
`rArr 2sin((2pi)/(n))cos((2pi)/(n))=sin((3pi)/(n))`
`sin ((4pi)/(n))=sin((3pi)/(n))`
`rArr (4pi)/(n)=pi-(3pi)/(n)`
`rArr (7pi)/(n)=pi`
`rArr n=7`
