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.

#Geometry #Parabola #Centroid #Intersection #Tangent

Easy Math Editor

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Comments

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.

Krishna Sharma - 6 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$