Two lines go from the apex of a triangle down to its base, and two more go from the bottom-left corner to the right side. How many triangles are there in all?

Using no inner line: 1. Using one: each of the 4 inner lines splits the triangle in 2, so 8. Using two: the two apex lines give 1, the two corner lines give 1, and each of the 4 crossing pairs gives 3, so 14. Using three: each corner line cuts a triangle out of the two apex lines, and each apex line cuts one out of the two corner lines, 4. Total $1+8+14+4=27$.