I have a set of questions I am struggling to comprehend.
The question is:
Points A, B, C and D are defined by position vectors a, b, c and d respectively. If $\overrightarrow{AB}$ + $\overrightarrow{CD}$ = $0$.
a) Express d in terms of a, b and c.
$d=a-b+c$
b) Show that AC and BD bisect each other.
Now here is where I'm struggling. I recognise:
$$\overrightarrow{AC}=c-a$$ $$\overrightarrow{BD}=d-b$$
and that
$$\overrightarrow{AB}=\overrightarrow{-CD}$$
but I can't figure out how to get them into a reasonable format to make the proof that they bisect each other. I'm assuming you use something like
$$\overrightarrow{OA} + \overrightarrow{AC} * \alpha = \overrightarrow{OB} + \overrightarrow{BD} * \beta $$
But it doesn't seem to work like this.
Edit:
I think it could also be done by showing that
$$\overrightarrow{OA} + 1/2* \overrightarrow{AC} = \overrightarrow{OB} + 1/2* \overrightarrow{BD}$$ and substituting in equivalent things like $$d=a-b+c$$ to solve it?