Gaps Between Rectangles Connected at the Center

Question

I'm experimenting with a vector based graphics style with objects represented as series of line segments with a given width(it would probably be easier to think of these as rectangles). The problem is that these segments are connected at the center and leave a gap (shown below). I've determined that the most efficient way to cover this gap is simply to cover it with a triangle, and since I'm working in OpenGL, all I need are the points of the two points that don't overlap with the other rectangle, the third point being the center point where the two line segments(rectangles) are connected. How can I determine which points I need to use for the triangle, given that I have all of the points from both rectangles?

EDIT: I will also accept alternative solutions, as long as they cover up that gap.

EDIT 2: Nevermind, I solved it. I'll post code once I have better Internet connection.

Answer 1

The solution I found assumes that there are 3 basic cases:

First, the three unique center points for the two rectangle proceed upward (positive y direction) so the gap is either on the left or right of the connection. In my code, I had the corner points of the rectangle organized by their orientation to the left or right of the center point, so if the bottom rectangle's left point is below the top rectangle's left point, then the gap is between the left points of the two rectangles, otherwise the gap is between the right points.

Second, the three unique center points have a maximum at the center most of the center points, so the gap is on the top. The gap is then between the two points with the maximum y values.

Third, the three unique center points have a minimum at the center most of the center points, so the gap is on the bottom. The gap is then between the two points with the minimum y values.

[I'll post pictures of the example cases if it is requested]