Tangents drawn from the point $\mathrm{O}$, meet the cirle at $\mathrm{P}$ & $\mathrm{Q}$ and $\mathrm{C}$ be the centre of given circle. Then points $\mathrm{O},\mathrm{ }\mathrm{P},\mathrm{ }\mathrm{C}$ and $\mathrm{Q}$ are concyclic. That means any circle passing through $\mathrm{O},\mathrm{ }\mathrm{P}$ and $\mathrm{Q}$ also passes through $\mathrm{C}$ and $\mathrm{O}\mathrm{C}$ is the diameter for this circle. Hence mid point of $\mathrm{O}\mathrm{C}$ is the circumcentre of triangle $\mathrm{O}\mathrm{P}\mathrm{Q}$.




Coordinates of circumcenter $=\left(\frac{-\mathrm{g}}{2},\frac{-\mathrm{f}}{2}\right)$