If the normals at $P, Q, R$ on the rectangular hyperbola  $xy=c^2$ intersect at a point S on the hyperbola, then  centroid of the triangle $PQR$ is at

Normal at any point $(ct,c/t)$ on the hyperbola $xy=c^2$  if $y-(c/t)=t^2(x-ct)\text{ or }{t}^{3}x-ty+c-c{t}^4=0$ If it passes through another points $(c\alpha ,c/\alpha )$ on the hyperbola, then 

$t^3\times c\alpha -t\times \frac{c}{\alpha }+c-c{t}^4=0$

$\begin{array}{l}\Rightarrow t^3{\alpha }^2-t+\alpha -\alpha t^4=0\\ \Rightarrow \left(t^3\alpha +1\right)(\alpha -t)=0\\ \Rightarrow t^3\alpha +1=0\text{ as }\alpha \ne t\end{array}$

which gives three values of t say $t_1,t_2,t_3$ and hence three points P, Q, R on the hyperbola the normals at  which pass through S.

We have $t_1+t_2+t_3=0=\mathrm{\Sigma }{t}_1{t}_2\text{ and }{t}_1{t}_2{t}_3=-1/\alpha$

Centroid of the triangle PQR is

$\left(\frac{c\left(t_1+t_2+t_3\right)}{3},\frac{c\left(1/t_1+1/t_2+1/t_3\right)}{3}\right)=(0,0)$

which is the centre of hyperbola.