Question 1154225: Help!!!!
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.
Link of shape: https://latex.artofproblemsolving.com/b/c/3/bc36d6a86ebdd87399c8b07d66ec053e8593c264.png
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
Straightforward, using midpoint formula and slope-intercept form of the equation of a line. But it is tedious; you need to do the work.
(1) Use the midpoint formula and the given coordinates of A, B, and C to find the midpoints D, E, and F.
(2) Use the coordinates of A and D to find the slope of AD; then use the slope and one of the points to determine the equation of AD. Repeat to find the equations of BE and CF.
(3) Solve the system of any two of the three equations to find their point of intersection; then show that that point satisfies the third equation.
I'll get you started....
D = midpoint of BC = ((-5+7)/2,(2-8)/2) = (1,-3)
slope of AD: (6-(-3))/(3-1) = 9/2
equation of AD:
y = mx+b
-3 = (9/2)(1)+b
-3 = 9/2+b
b = -15/2
y = (9/2)x-15/2
Use the same process to find the equations of BE and CF; then proceed as outlined above.
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