2016/042) If $(m_r,\frac{1}{m_r})$ for r from 1 to 4 are 4 points that lie on a circle show that $m_1m_2m_3m_4= 1$

we have general equation of a circle
$x^2+y^2+ 2gx + 2fy + c = 0$
let a point $p,\frac{1}{p}$ be on the circle
then we get
$p^2 + \frac{1}{p^2} + 2gp + \frac{2f}{p} + c = 0$
or $p^4 + 1 + 2gp^3 + 2fp + cp^2=0$
or $p^4 + 2gp^3 + cp^2 + 2fp + 1=0$
the 4 roots are $m_1,m_2,m_3,m_4$ and hence product of roots = $m_1m_2m_3m_4=1$ (the constant term)