All Hail Euler the ruler!

Geometry Level 4

Consider a triangle $\Delta ABC$ whose circumcenter is at the origin. If in $\Delta ABC$ , the coordinates of the centroid $G$ are $\left(x_G,y_G\right)$ and the coordinates of the orthocenter $H$ are $\left(x_H,y_H\right)$ .

Find the ratio $\dfrac{x_Hy_H}{x_Gy_G}$ .

Bonus: Don't forget the lonely incenter!