All Hail Euler the ruler!

Geometry Level 4

Consider a triangleΔABC\Delta ABC whose circumcenter is at the origin. If in ΔABC\Delta ABC, the coordinates of the centroid GG are (xG,yG)\left(x_G,y_G\right) and the coordinates of the orthocenter HH are (xH,yH)\left(x_H,y_H\right).

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

Bonus: Don't forget the lonely incenter!

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