# Center

In the triangle ABC is point D[1,-2,6], which is the center of the |BC| and point G[8,1,-3], which is the center of gravity of the triangle.

Find the coordinates of the vertex A[x,y,z].

Result

x =  22
y =  7
z =  -21

#### Solution:

$AD: X = G + t(D-G) \ \\ \ \\ x = 8 -7t \ \\ y= 1 -3t \ \\ z= -3 +9t \ \\ \ \\ |AG|:|GD| = 2:1 => t = -2 \ \\ D ... t=1 \ \\ G ... t=0 \ \\ A ... t=-2 \ \\ \ \\ x_A = 22$
$y_A = 7$
$z_A = -21$

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