Given are $\Delta ABC$ with $A(-2,2); B(6,0); C(2,-4)$
a) Prove analytically that the medians of $\Delta ABC$ go through 1 point and look for the coordinates of the centroid $Z$.
Side question How does one get equations of the medians?
I've tried calculating the median that goes through B but I get the negation of what it should be i.e. $y=\frac{-1}{6}x+1$
As you see in figure:
$$x_G=\frac{x_A+x_B+x_C}3=2$$
$$y_G=\frac{y_A+y_B+y_C}3=-0.66$$
Find coordinates of midpoints of the sides. The coordinates of G must satisfy equation of lines from each vertex and midpoint of it's opposite side.
You can also find the mirror of each vertex about midpoint of opposite side. As can be seen in figure the mirrors of two vertices must be col-linear with third vertex which is easy to check.This is possible only if all medians go through one point( a property of medians).