What is the maximum number of points at which the sides of a triangle can intersect the sides of another triangle?

The Wording of the question leaves room for two interpretations, but I believe the expected answer is six. Think of the "Star of David" (also known as the Magen David). See the "Web Link" at the bottom left. But that's for only one orientation. There are an infinite number of points on each line segment, so if you rotate one triangle with respect to the other, different pairs of points will be involved.