Find the number of triangles in the given figure.

To count every triangle in a figure built from several intersecting straight lines, classify each triangle by how many of the smallest (undivided) regions it is made up of — call this its order. List the order-1 (simplest) triangles first, then find every order-2 triangle formed by merging two adjoining order-1 regions along a shared edge, then order-3, and so on, and add the counts for every order without missing or repeating a combination.

The figure may be labelled as shown, with F and E marking the two points where the drawn lines cross inside the rectangle ABCD.

1. Order-1 (simplest) triangles — each made of a single undivided region: AFB, FEB, EBC, DEC, DFE and AFD, i.e. 6 triangles.

2. Order-2 triangles — each made by merging two adjoining order-1 regions: AEB, FBC, DFC, ADE, DBE and ABD, i.e. 6 triangles.

3. Order-3 triangles — each made by merging three adjoining regions: ADC and ABC, i.e. 2 triangles.

4. Order-4 triangle — the single largest triangle made by merging all four regions on that side: DBC, i.e. 1 triangle.

Adding every order: 6 + 6 + 2 + 1 = 15 triangles in the figure.

Cross-check: these four triangles, each formed directly from three of the rectangle's own corners — ABD, ABC, ADC and BCD — appear exactly once within the levels counted above (ABD among the two-region triangles, ABC and ADC among the three-region triangles, and BCD as the single four-region triangle). This is a necessary consistency check on the level count above; the completeness of the full list of 15 rests on having examined every combination of the six atomic regions systematically, level by level, as done above.