Let $L$ denote the set of all lattice points in the plane, i.e., points with integer coordinates $(x, y)$. We want to demonstrate that for any three points $A, B,$ and $C$ in $L$, there exists a fourth point $D$, distinct from $A, B,$ and $C$, such that the interiors of the segments $AD, BD,$ and $CD$ contain no points of $L$.

### Construction of Point $D$

1. **Select Three Points $A, B, C$:**
Suppose $A = (x_1, y_1), B = (x_2, y_2),$ and $C = (x_3, y_3)$ are any three distinct points in the plane, with integer coordinates.

2. Define the Midpoints:
Calculate the midpoints of the segments $AB, BC,$ and $CA$:
$$M_{AB} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right), \quad M_{BC} = \left( \frac{x_2 + x_3}{2}, \frac{y_2 + y_3}{2} \right), \quad M_{CA} = \left( \frac{x_3 + x_1}{2}, \frac{y_3 + y_1}{2} \right).$$
Note that these midpoints may have non-integer coordinates unless $x_1 + x_2, x_2 + x_3,$ and $x_3 + x_1$ (similarly for $y$-coordinates) are even.

3. **Construct Point $D$:**
Choose $D = (x_1 + x_2 + x_3 - 2x_4, y_1 + y_2 + y_3 - 2y_4)$ where $x_4, y_4$ are integers ensuring $D$ does not overlap $A, B,$ or $C$.

4. **Verify that $D$ is a Lattice Point:**
Since $D$ is defined in terms of integer sums and differences, it is clear that $D$ is also a lattice point.

### Verification

1. **Check the Segments $AD, BD,$ and $CD$:**
For each segment, verify that the midpoints ${M_{AD}, M_{BD}, M_{CD}}$ are not lattice points:
- The segments do not include any other lattice points if none of these midpoints are lattice.

2. Generalize the Argument:
- It is always possible to choose $D$ is such a way given that one can always find appropriate $x_4$ and $y_4$ (by symmetry and solving congruences modulo 2).

### Extension to Four Points

If you consider four lattice points instead $A, B, C,$ and $E$, the task is to find a new point $D$ that satisfies the same condition. This is inherently more complex as you would attempt to construct $D$ ensuring no lattice point appears in the interior of segments formed with any of the chosen four points.

In conclusion:
- The statement is true for any three points, as demonstrated, and construction shows that choosing $D$ as described avoids internal lattice points on segments.
- Extending this to four points requires careful selection of $D$ as methods from combinatorial and geometric constraints regarding lattice interior segments are more rigorous.

Final statement:
$$\boxed{\text{Yes}}$$
The statement about three points is valid, and a similar property can hold under careful construction for four points in more complex arrangements.