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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.

 

https://latex.artofproblemsolving.com/b/c/3/bc36d6a86ebdd87399c8b07d66ec053e8593c264.png

 

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.

 Apr 23, 2021
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Median from A: y = 6x - 12

Median from B: y = -2x/7 + 4/7

Median from C: y = -8x/5 + 16/5

 

All three medians pass through (2,0)

 Apr 24, 2021

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