Question

ABC is a triangle, D is the mid-point of BC. Prove that $\overrightarrow{AB}+\overrightarrow{AC}=2\overrightarrow{AD}$.

Answer 1

According to this figure and based on triangle law of vectors we can write following equations:

$$\overrightarrow{AB}+\overrightarrow{BD}=\overrightarrow{AD}$$

$$\overrightarrow{AC}=\overrightarrow{AD}+\overrightarrow{DC}$$

Summing the equations:

$$\overrightarrow{AB}+\overrightarrow{AC}+\overrightarrow{BD}=2\overrightarrow{AD}+\overrightarrow{DC}$$........(1)

Since, $D$ is the mid point $\frac{\overrightarrow{BC}}{2}=\overrightarrow{BD}=\overrightarrow{DC}$

Equations (1) $⟹\overrightarrow{AB}+\overrightarrow{AC}=2\overrightarrow{AD}$

Hence Proved,