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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