Question 82
A median of a triangle is the segment that connects a vertex to the midpoint of the opposite side. Consider the triangle $ABC$ in the following figure and be $M$, $N$ e $P$ the midpoints of the sides $BC$, $CA$ e $AB$, respectively. the medians $BN$ e $CP$ cut to the point $G$. Be $X$ the midpoint of the segment $AG$.
(a) Using the quadrilateral $GPXN$, check that the point $G$ divide the segment $CP$ in reason $2:1$, that is, that $CG=2 \cdot GP$.
(b) From the previous item, check that the median $AM$ cut the median $CP$ at the same point $G$. Note that this shows that the three medians of a triangle pass through the same point. This point is called the barycenter of the triangle.
(c) Suppose the medians $BN$ e $CP$ have the same length, check that $AC=AB$.
(a) Using the quadrilateral $GPXN$, check that the point $G$ divide the segment $CP$ in reason $2:1$, that is, that $CG=2 \cdot GP$.
(b) From the previous item, check that the median $AM$ cut the median $CP$ at the same point $G$. Note that this shows that the three medians of a triangle pass through the same point. This point is called the barycenter of the triangle.
(c) Suppose the medians $BN$ e $CP$ have the same length, check that $AC=AB$.