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 $A = (5,4)$, $B = (-9, 6)$, and $C = (1, -4)$. (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.

Guest Apr 22, 2023