a theoretical question regarding parallelogram and ratio of intersection of a line segment in it by a diagonal

Question

Suppose $ABCD$ is a parallelogram. AC and BD are diagonals. they intersect each other at point O. P is the midpoint of AO. E is the midpoint of BC. Does the other diagonal (BD) intersect PE into ratio 1:1?

Answer 1

Yes. Put $A$ at the origin of the coordinate system and let $u=\overrightarrow {AD}$ and $v=\overrightarrow {AB}$. Then the midpoint of $[PE]$ is $\frac{3u+5v}{8}$. This can be rewritten as $u+\frac{5}{8}(v-u)$, which is indeed on the line $(BD)$, since that line is exactly all of the points of the form $u+t(v-u)$, for any real number $t$.