Find the number of triangles in the given figure.

To count every triangle in a composite figure, first list all the smallest (non-decomposable) triangles created by the drawn lines. Then check every way two or more of those smallest pieces join together, keeping only the combinations whose outer boundary itself has exactly three straight sides — a triangle. Add these to the smallest-triangle count, taking care never to count the same triangle twice.

1. In triangle ABC, D lies on AB, E lies on AC, and F lies on BC, with DE, EF and FD drawn — this creates four smallest triangles: ADE, DBF, DEF and EFC.

2. Combining any two adjacent smallest triangles (for example ADE with DEF, or DBF with DEF) produces a four-sided region, not a triangle, so none of these two-piece combinations qualify.

3. Combining any three of the four smallest triangles likewise produces a boundary with more than three sides, so no three-piece combination qualifies either.

4. Combining all four smallest triangles together exactly reconstructs the full outer triangle ABC, whose boundary has exactly three sides — this is one further valid triangle.

Cross-check by naming every triangle found: ADE, DBF, DEF, EFC and ABC — five distinct triangles in total, confirming the count of 4 (smallest) + 1 (outer) = 5.