If $\left(m_i, 1/m_i\right), i=1, 2, 3, 4$ are concyclic points, then the value of $m_1 m_2 m_3 m_4 is..................$

Let the equation of a circle be $x^2+y^2+2gx+2fy+c=0$

$\left(m, \frac{1}{m}\right)$ is a point on the circle $\Rightarrow$$m^2+\frac{1}{m^2}+2gm+2f\left(\frac{1}{m}\right)+c=0$

$\Rightarrow m^4+1+2g{m}^3+2fm+c{m}^2=0\Rightarrow m^4+2g{m}^3+c{m}^2+2fm+1=0$

Let $m_1, m_2, m_3, m_4$be the roots of this equation.

Then $\left(m_i, \frac{1}{m_i}\right), i=1, 2, 3, 4$ one con cyclic.    Product of the roots ; $m_1{m}_2 m_3{m}_4 =1$.