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Point $A, B, C, D$ have position vectors $a, b, c, d$ respectively relative to an origin O.

If $P$ divides $AB$ in the ratio $1:2$ and $Q$ divides $CD$ in the ratio $1:2$, obtain an expression for the position vector of $X$, where $X$ is the midpoint of $PQ$.

If $ABCD$ is a parallelogram show that $X$ is the point in which the diagonals $AC$ and $BD$ intersect.

I did the following diagram to work out this problem.

Diagram

I figured that to get $\overrightarrow{OX}$, I need $\overrightarrow{PQ}$, $\overrightarrow{PX}$ and $\overrightarrow{AP}$, which I got as below,

$ \begin{align} \overrightarrow{PQ} &= \overrightarrow{PA} + \overrightarrow{AD} + \overrightarrow{DQ} \\ &= -\frac{2}{3} a - \frac{1}{3} b + \frac{2}{3} c + \frac{1}{3} d \\ \\ \overrightarrow{PX} &= \frac{1}{2}\overrightarrow{PQ} \\ &= -\frac{1}{3} a - \frac{1}{6} b + \frac{1}{3} c + \frac{1}{6} d \\ \\ \overrightarrow{AP} + \overrightarrow{PX} &= \overrightarrow{AX} \\ \overrightarrow{AX} &= \frac{1}{3} \overrightarrow{AB} + \overrightarrow{PX} \\ &= -\frac{2}{3}a + \frac{1}{6}b + \frac{1}{3}c + \frac{1}{6}d \\ \\ \overrightarrow{OX} &= \overrightarrow{OA} + \overrightarrow{AX} \\ &= \frac{1}{3}a + \frac{1}{6}b + \frac{1}{3}c + \frac{1}{6}d \\ &= \frac16(2a+b+2c+d) \end{align} $

For the second part I figure I need to show that $X$ lies on $AC$ and $BD$. ie:- $\overrightarrow{AX} = k\overrightarrow{AC} |k\overrightarrow{XC}$ and $\overrightarrow{BX} = k\overrightarrow{BC} |k\overrightarrow{XD}$.

I tried doing this but end up with vectors without all the components. So I am unable to represent the vectors as multiples of each other. What am I missing? How do you solve the last part of this question?

Thanks for all your help!

1 Answers 1

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In a parallelogram, you have $b-a=c-d$, so $b-a+d-c=0$. Adding one twelfth of that to your last displayed equation yields $\overrightarrow{OX}=\frac14(a+b+c+d)$, which is the midpoint of the parallelogram and the point where the diagonals intersect.

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    I see what you mean about the one twelfths now. I had interpreted it as add one twelfth + $\overrightarrow{OX}$ instead of add both the equations together which makes more sense now.2011-07-22