Proof for triangle division into sevenths
Basically, what's going on is that the division along the side of the triangle serves as the necessary references for a larger triangle. So this is an exact construction for division into any number 2n-1.
*edit* In case it isn't obvious, I put in a dotted line showing why it's 2n-1 and not some other number. My brother also pointed out that this will work for ANY triangle, not just an equilateral one. Following up on that, it seems that this triangle is an application of the Crossing Diagonals method on half a parallelogram.