Hello, darksoulzero!

$\displaystyle \text{Triangle }D{E}F\text{ has vertices: }D(1,3)\text{ and }E(6,1)\text{, and centroid at }C(3,4).$

$\displaystyle \text{Determine the coordinates of point }F.$

Code:

1
F o-----+
\ :
\ :4
\ :
\ :
\:0.5
(3,4)o---+
D C\ :
o \ :2
(1,3) * \:
M o
(3.5,2) * E
o
(6,1)

We have vertices $\displaystyle D(1,3)$ and $\displaystyle E(6,1)$, and centroid $\displaystyle C(3,4).$

The midpoint of $\displaystyle DE$ is: $\displaystyle M(3\tfrac{1}{2},\,2).$

The median to side $\displaystyle DE$ starts at $\displaystyle \,M$, passes through $\displaystyle \,C,$

. . and extends to $\displaystyle \,F$, where: .$\displaystyle FC \,=\,2\!\cdot\!CM.$

Going from $\displaystyle \,M$ to $\displaystyle \,C$, we move up 2 and left $\displaystyle \frac{1}{2}$

Hence, going from $\displaystyle \,C$ to $\displaystyle \,F$, we move up 4 and left 1.

Therefore, we have: .$\displaystyle F(2,8).$