Q:

Can the intersection of triangles produce a parallelogram?

I came across a question asking for a list of figures  that can be formed by the intersection of two arbitrary triangles. Attempts showed that we can get at least, lines, rectangles, squares, and triangles of course. I also found something I think is not on the list. Intuitively it looks like:

Observation. a parallelogram cannot be obtained by the intersection of two triangles

since the sides never meet at a point to determine a triangle. How can I go about proving this?

A:

See below. The blue triangle was formed by rotating the red one 180 degrees, and then shifting. The purple area is a parallelogram.

