A median of a triangle is a line segment joining a vertex of a triangle to the midpoint of the opposite side. The three medians of a triangle are drawn below.

[asy]
pair A, B,C, X,Y, Z;
A = (0,0);
B = (1,0);
C = (0.3,0.8);
X = (B+C)/2;
Y = (A+C)/2;
Z = (A+B)/2;

draw(A--X, red);
draw(B--Y, red);
draw(C--Z, red);
draw(A--B--C--A);
[/asy]

Note that the three medians appear to intersect at the same point! Let's try this out with a particular triangle. Consider the triangle $ABC$ with $A = (3,6)$, $B = (-5,2)$, and $C = (7,-8)$.

(a) Let $D,$ $E,$ $F$ be the midpoints of $\overline{BC},$ $\overline{AC},$ $\overline{AB},$ respectively. Find the equations of medians $\overline{AD},$ $\overline{BE},$ and $\overline{CF}.$

(b) Show that the three medians in part (a) all pass through the same point.