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# Centroids of Triangles on a Parabola

The parabola $$y=x^2$$ has three points $$P_1,P_2,P_3$$ on it. The lines tangent to the parabola at $$P_1, P_2, P_3$$ intersect each other pairwise at $$X_1,X_2,X_3$$. Let the centroids of $$\triangle P_1P_2P_3$$ and $$\triangle X_1X_2X_3$$ be $$G_P, G_X$$ respectively. Prove that $$G_PG_X$$ is parallel to the y-axis.

Note by Daniel Liu
2 years, 9 months ago

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General parametric coordinates of $$x^2 = 4ay$$ is

$$(2at, at^2)$$

And point of intersection of tangents from $$t_1, t_2$$ is

$$(a(t_1 + t_2), at_1t_2)$$

Let P,Q,R be $$t_1,t_2,t_3$$ resp.

We just need to check the x-coordinate of the Centroids are same or not

x-coordinate of Centroid of PQR is $$(\dfrac{2a(t_1 + t_2 + t_3)}{3})$$

x- coordinates of X,Y,Z are $$a(t_1 + t_2), a(t_2+ t_3), a(t_3 + t_1)$$

x- coordinate of Centroid of XYZ is

$$\dfrac{2a(t_1 + t_2 + t_3)}{3}$$

Hence Proved.

I m just lazy to prove point of intersection of tangents, I'll do if you want the proof.

- 2 years, 9 months ago