A rectangle stands on the base of a triangle with its top corners on the two sides, and the apex is joined to the rectangle's bottom corners. How many triangles are in the figure?

From the apex four lines go down. The rectangle's top edge cuts all four: $\left(\genfrac{}{}{0ex}{}{4}{2}\right)=6$. The base cuts all four: 6 more. On each side the apex, a top corner and a bottom corner of the rectangle make 1; inside the rectangle each top corner gives 1; outside it each bottom corner gives 1: $2+2+2=6$. Total $6+6+6=18$.