Consider a regular pentagon $ABCDE$. All five of its internal diagonals ($AC, AD, BD, BE, CE$) are drawn, forming an inner star (pentagram) enclosing an inverted central miniature pentagon. How many total triangles (of all sizes) are formed within this geometric figure? ________ (answer in integer).
Answer Key:
35
Step-by-Step Solution & Explanation
**Method 1: Geometric Size Classification**
1. Outer small triangles (star points formed by 1 outer edge and 2 diagonal segments): $5$
2. Small triangles sharing an edge of the central inner pentagon: $5$
3. Intermediate triangles formed by 2 outer vertices and 1 inner diagonal intersection: $10$
4. Large triangles formed with 1 outer edge and 2 full diagonals: $5$
5. Largest triangles with all 3 vertices on the outer pentagon: $\binom{5}{3} = 10$
$\text{Total Triangles} = 5 + 5 + 10 + 5 + 10 = 35$
Integer answer: 35.