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].

Correct result:

x =  22
y =  7
z =  -21

Solution:

$$\begin{gathered} AD:X=G+t(D-G) \\ x=8-7t \\ y=1-3t \\ z=-3+9t \\ \mathrm{\mid }AG\mathrm{\mid }:\mathrm{\mid }GD\mathrm{\mid }=2:1=>t=-2 \\ D\mathrm{.}\mathrm{.}\mathrm{.}t=1 \\ G\mathrm{.}\mathrm{.}\mathrm{.}t=0 \\ A\mathrm{.}\mathrm{.}\mathrm{.}t=-2 \\ {x}_A=22 \end{gathered}$$

$y_A=7$

$z_A=-21$

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