## Problem on geometric proofs with vectors

Show that the line connecting the midpoints of two sides of a triangle is parallel to and half the length of the third side.
• ## Solution

Next, we express the two midpoints in terms of $\bfa, \bfb,\bfc$.
Recall that
Similarly, the other midpoint depicted is $\frac{\bfb + \bfc}{2}$:
We are trying to compare the line segements from $\bfa$ to $\bfb$ and from $\frac{\bfa+\bfc}{2}$ and $\frac{\bfb+\bfc}{2}$. We now write both as vectors.
Recall that
Similarly, the vector from $\frac{\bfa+\bfc}{2}$ and $\frac{\bfb+\bfc}{2}$ is $$\frac{\bfb+\bfc}{2} - \frac{\bfa+\bfc}{2} = \frac{\bfb - \bfa}{2}$$
We conclude that the line segment connecting the midpoints of two sides of a triangle is parallel and half the length of the third side.