The midpoint between u and v is (u + v)/2. Convince yourself by drawing a parallelogram. This fact can be used to find conditions on c, d such that the linear combinations cu + dv fill in the line segment between u and v.
A better way to think about this: The vector from u to v is v - u. Starting at u, we can vary the length of v - u to fill in the line segment. Consider u + t(v - u) for t in [0, 1].
To fill in the entire triangle including its inside, use the same sort of reasoning: Starting at u, we can vary the lengths of v - u and w - u under certain restrictions.
