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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)

(a)

Let  D, E, F be the midpoints of  $\overline{BC},$ ,  \overline{AC},$ and $\overline{AB},$  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.

 Mar 21, 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)

 Mar 21, 2021

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