Find the coordinates of the centroid of the triangle whose vertices are $(x_1, y_1, z_1)$, $(x_2, y_2, z_2)$ and $(x_3, y_3, z_3)$.

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Accepted Answer

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> [!TLDR]
> Using the fact that a median is divided by the centroid in the ratio $2:1$, and applying the section formula, gives the centroid as the average of the three vertices' coordinates.
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> Let the triangle have vertices $A(x_1,y_1,z_1)$, $B(x_2,y_2,z_2)$, $C(x_3,y_3,z_3)$.
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> Let $D$ be the midpoint of side $BC$:
> $$D = \left(\frac{x_2+x_3}{2}, \frac{y_2+y_3}{2}, \frac{z_2+z_3}{2}\right)$$
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> $AD$ is a median of the triangle. The centroid $G$ divides each median in the ratio $2:1$ from the vertex, i.e. $AG:GD = 2:1$.
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> By the section formula, the point dividing the segment from $A(x_1,y_1,z_1)$ to $D$ in the ratio $2:1$ is: …

Plane	Equation	Description
$xy$ -plane	$z = 0$	the floor — all points with zero height
$yz$ -plane	$x = 0$	one wall — all points with zero $x$
$zx$ -plane	$y = 0$	the other wall — all points with zero $y$

Q.Find the coordinates of the centroid of the triangle whose vertices are $(x_1, y_1, z_1)$ , $(x_2, y_2, z_2)$ and $(x_3, y_3, z_3)$ .

Concept understanding — 3D Coordinate Geometry

3D Coordinate Geometry

The Intuition: Three Numbers, One Point

The Precise Statement

How to Read a 3D Point

The Three Coordinate Planes

Distance Between Two Points

Section Formula (Internal Division)

Q.Find the coordinates of the centroid of the triangle whose vertices are (x1,y1,z1)(x_1, y_1, z_1)(x1​,y1​,z1​), (x2,y2,z2)(x_2, y_2, z_2)(x2​,y2​,z2​) and (x3,y3,z3)(x_3, y_3, z_3)(x3​,y3​,z3​).

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Q.Find the coordinates of the centroid of the triangle whose vertices are $(x_1, y_1, z_1)$ , $(x_2, y_2, z_2)$ and $(x_3, y_3, z_3)$ .

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