Serious geometry question?

Question

Regional parts

Each vertex of a triangle is joined by straight line to six points on the opposite side of the triangle. no three of the joining lines pass through the same point. Into how many regions do these 18 lines divide the interior of the triangle

Answer 1

That's interesting, the formula is 3n(n+1)+1, where n is the number of lines drawn from each vertex. So, for n = 6, we have 127.

The formula is derived in the following way: First, with n lines from vertex 1 and 2, the triangle is divided into n² regions. Then n lines from vertex 3, drawn closely to one side of the triangle, add n(n+1) and n² more regions, for a total of 3n² + 3n + 1 regions. Sweeping any one line across the triangle leaves the total number intact, as whenever it "leaves a region", it enters another. So, it does not matter where the 3n lines are drawn, so long they are straight.

Answer 2

I drew it out and found 108. There must a a mathematical way to calculate this (maybe something to do with networks, nodes, pathways).