Point $M$ is the midpoint of side $BC$ of triangle $ABC$. Prove that $\vec{AB} + \vec{AC} = 2\vec{AM}$.

1. Problem Description

Point $M$ is the midpoint of side $BC$ of triangle $ABC$. Prove that $\vec{AB} + \vec{AC} = 2\vec{AM}$.

2. Solution Steps

We are given that $M$ is the midpoint of $BC$. This means $\vec{BM} = \vec{MC}$.

We can express $\vec{AM}$ as $\vec{AB} + \vec{BM}$ and $\vec{AM}$ as $\vec{AC} + \vec{CM}$.

Therefore, $\vec{AM} = \vec{AB} + \vec{BM}$ and $\vec{AM} = \vec{AC} + \vec{CM}$.

Adding these two equations, we get

$2\vec{AM} = \vec{AB} + \vec{AC} + \vec{BM} + \vec{CM}$.

Since $\vec{BM} = \vec{MC}$, we have $\vec{CM} = -\vec{MC} = -\vec{BM}$.

Therefore, $\vec{BM} + \vec{CM} = \vec{BM} - \vec{BM} = \vec{0}$.

So, $2\vec{AM} = \vec{AB} + \vec{AC} + \vec{0}$.

Hence, $2\vec{AM} = \vec{AB} + \vec{AC}$.

3. Final Answer

$\vec{AB} + \vec{AC} = 2\vec{AM}$