Given D is the midpoint of AB and E is the midpoint of AC. If AD = a and AE = b, express AB, AC, DE, and BC in terms of a and b.

Since D is the midpoint of AB, the vector $\vec{AB}$ is twice the vector $\vec{AD}$.

$$\vec{AB} = 2\vec{AD}$$

$$\vec{AB} = 2\mathbf{a}$$

The vector $\vec{AB}$ is $\boxed{2\mathbf{a}}$.

Since E is the midpoint of AC, the vector $\vec{AC}$ is twice the vector $\vec{AE}$.

$$\vec{AC} = 2\vec{AE}$$

$$\vec{AC} = 2\mathbf{b}$$

The vector $\vec{AC}$ is $\boxed{2\mathbf{b}}$.

To go from D to E, we can go from D to A, then from A to E.

$$\vec{DE} = \vec{DA} + \vec{AE}$$

Since $\vec{DA}$ is the reverse of $\vec{AD}$, $\vec{DA} = -\vec{AD} = -\mathbf{a}$.

$$\vec{DE} = -\mathbf{a} + \mathbf{b}$$

The vector $\vec{DE}$ is $\boxed{\mathbf{b} - \mathbf{a}}$.

To go from B to C, we can go from B to A, then from A to C.

$$\vec{BC} = \vec{BA} + \vec{AC}$$

From part (a), $\vec{AB} = 2\mathbf{a}$, so $\vec{BA} = -\vec{AB} = -2\mathbf{a}$.
From part (b), $\vec{AC} = 2\mathbf{b}$.

$$\vec{BC} = -2\mathbf{a} + 2\mathbf{b}$$

$$\vec{BC} = 2(\mathbf{b} - \mathbf{a})$$

The vector $\vec{BC}$ is $\boxed{2(\mathbf{b} - \mathbf{a})}$.