Midpoint of a segment

Question

Let ABC be a triangle, I and J are two points such that : $\overrightarrow{AI}=\frac {2}{3}\overrightarrow {AB}$ and $\overrightarrow{AC}=\frac {1}{2}\overrightarrow {AJ}$.

(BC) and (IJ) intersect in O.

Show that O is the midpoint of the segment [BC].

Answer 1

HINT

We have that

• line $IJ:\quad P(t)=AI+t(AJ-AI)$
• line $BC:\quad Q(s)=AB+s(AC-AB)$

then equate $P(t)=Q(s)$ and use that ${AI}=\frac {2}{3} {AB}$ and ${AC}=\frac {1}{2} {AJ}$ to find the intersection point $O$. Finally check that $O$ is the midpoint of $BC$.