I've come up with a theory for how to solve it.

Each dot has two places it can go to (with a few exceptions that only have one). So that dot's total number of options will be the sum of the total number of options from each of the two places it can go. So, for example, the entire right row will only have 1 option (to go up) Then starting at the next row (remember to start at the point (5,5) since you cannot cross that line. (5,5) can only go to the right, so it is the sum of 1, and has 1 option. The dot below it can go up or right, so it is the sum of 1 and 1 = 2. The next will be 2+1=3, and so on. When you get to the origin, you should have the sum of the total number of paths unless I messed up. Then you multiply it by two, because it is two triangles.

I'll work on a diagram.

Note that this may be wrong, but I think it will work.

edit: I attached my unverified solution. For one triangle, there are 132 possibilities, so for the whole thing, there are 264 possibilities.