Demonstrate vectors equality

Question

So, I have a quadrilateral ABCD, E and F are the midpoints of the diagonals and I have to demonstrate that:

$$\overrightarrow{EF} =\frac{1}{2} \left(\overrightarrow{AB} +\overrightarrow{CD}\right) =\frac{1}{2} \left(\overrightarrow{AD} +\overrightarrow{CB}\right)$$

I just started to learn this stuff. I know just basic vectors arithmetic and I have no clue how to solve this.

Answer 1

We need simply to show that given $\vec R$ and $\vec Q$ the midpoint $\vec M$ is given by

$$\vec M = \frac{\vec R + \vec Q}{2}$$

then consider the parametric equation for the line $\vec {RQ}$ that is

$$\vec P(t)=\vec R+t(\vec Q-\vec R)$$

and note that

• $t=0 \implies \vec P(0)=\vec R$
• $t=1 \implies \vec P(1)=\vec Q$
• $t=\frac12 \implies \vec P(\frac12)=\vec M=\vec R+\frac12(\vec Q-\vec R)=\frac{\vec R + \vec Q}{2}$