6. (3 points) A quadrilateral is divided into 1000 triangles. What is the maximum number of different points at which the vertices of these triangles can be located?
Answer: 1002

Solution: The sum of the angles of a quadrilateral is $360^{\circ}$, and for thousands of triangles, it is $180000^{\circ}$. Where did the extra $1796400^{\circ}$ come from? Each internal vertex, where only the angles of triangles meet, adds $360^{\circ}$. Each vertex on the side of the original triangle or on the side of one of the triangles in the partition adds $180^{\circ}$. To achieve the maximum, it is necessary to have only points of the second type, and there should be $179640: 180=998$ of them. The example is constructed by sequentially adding new points to the boundaries of the original quadrilateral and the previously constructed triangles.

Adding the three vertices of the original quadrilateral, we get the answer 1002