Find the number of triangles in the given figure.

Concept: In a compound figure built from intersecting lines, count triangles level by level — first every "simplest" triangle (one not cut by any further line), then every larger triangle formed by combining 2, 3, 4 or more of those simplest triangles, treating each combination level separately — and only then add every level together. Stopping the count early, before the largest composite triangles are included, is the most common mistake in this type of question.

Applying it to this figure: Label the square ABDF, with C marked on side BD and E marked on side FD. Diagonals AD and BF are drawn corner to corner, and the lines GC (horizontal), GE (vertical) and EC (diagonal) are also drawn, creating the interior points G, H, I, J and K shown below.

1. Simplest (undivided) triangles: EFH, BIC, GHJ, GIJ, EKD and CKD — 6 triangles.

2. Triangles formed from two simplest triangles combined: ABJ, AFJ, GCK, GEK, CED and GHI — 6 triangles.

3. Triangles formed from three components: GCD, GED, DJB and DJF — 4 triangles.

4. Triangles formed from four components: ABF and GCE — 2 triangles.

5. Triangles formed from five components: ABD and AFD — 2 triangles.

6. The single largest triangle, formed from all six components: FBD — 1 triangle.

7. Total = 6 + 6 + 4 + 2 + 2 + 1 = 21.

Cross-check: Adding only the first three levels gives 6 + 6 + 4 = 16, and adding the first four levels gives 6 + 6 + 4 + 2 = 18 — both are partial totals that stop the count early. The figure has 21 triangles once every level, through the largest triangle FBD, is included.