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!

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