Find the number of triangles in the given figure.

Concept

A figure built from a square, its diagonals, and an inscribed (rotated) square is counted by first labelling every point where lines cross, then classifying every triangle by how many of the smallest, indivisible regions it is built from — the single-region (simplest) triangles first, then the ones made of two adjacent regions, then any larger combinations — and finally adding every category together. Counting layer by layer like this stops composite triangles from being missed or counted twice.

Application

Label the figure as shown:

1. Simplest (single-region) triangles: IJO, BCJ, CDK, KQL, MLQ, GFM, GHN and NIO — 8 triangles.

2. Triangles made of two adjacent simplest regions each: ABO, AHO, NIJ, IGP, ICP, DEQ, FEQ, KLM, LCP and LGP — 10 triangles.

3. Triangles made of four simplest regions together: HAB, DEF, LGI, GIC, ICL and GLC — 6 triangles.

4. Add every category: 8 + 10 + 6 = 24.

Cross-check

Scan the figure once more to confirm none of the three groups overlaps and none was skipped — every named triangle has exactly three straight sides, and each is counted in exactly one of the three groups, so the running total can be trusted.

Hence, the total number of triangles in the figure is 24.