I'm working on a proof that looks like this:
Let $n$ be a positive integer. Given an equilateral triangle, place $n$ points on each side, dividing the side into $n+1$ equal segments.
Use the points to draw $n$ line segments parallel to each side of the triangle ($3n$ line segments in all).
Prove by induction that this will always divide the triangle into exactly $(n+1)^2$ little equilateral triangles.
But I am unsure of how to proceed past the base case.