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The problem presents an equilateral triangle \( PQR \) with side-length 1. Several lines trisect the angles of the triangle, forming a smaller equilateral triangle \( STU \). We are tasked with finding the side-length of this smaller equilateral triangle \( STU \).

### Given Information:
1. Triangle \( PQR \) is equilateral with side-length 1.
2. The lines \( PT \) and \( PU \) trisect the angle \( \angle RPQ \).
3. The lines \( RS \) and \( RT \) trisect the angle \( \angle QRP \).
4. The lines \( QS \) and \( QU \) trisect the angle \( \angle PQR \).
5. We need to determine the side-length of triangle \( STU \).

### Solution:
The key observation is that since the smaller triangle \( STU \) is also equilateral, the geometry of the trisectors suggests that this smaller triangle is similar to the larger triangle \( PQR \), but scaled down by a factor related to the angles created by the trisectors.

Using trigonometric properties of trisected angles, the side-length of \( STU \) is connected to the cosine of certain angles (like 20°, 80°, etc.). Based on the options provided, this involves cosine ratios for specific angles, typically found in triangle division problems involving angle trisectors.

### Examining the answer options:
Given the symmetry and geometric relations between the angles and sides, the correct formula for the side-length of \( STU \) would likely involve trigonometric expressions based on the angle trisectors.

**Answer: \( \boxed{\frac{1}{2}} \)** corresponds to the case where the smaller triangle's side-length is half of the original triangle's side-length.

Would you like to see a detailed breakdown of how the cosine ratios come into play, or do you have any specific questions about the steps involved?

### Follow-up questions:
1. How do trisectors of angles in an equilateral triangle create similar smaller triangles?
2. What role does the cosine function play in determining the side-length of similar triangles?
3. How are cosine values for specific angles like 20° and 80° calculated geometrically?
4. Can you describe the geometric relationship between the original triangle and the smaller one formed by trisectors?
5. Why is the side-length of \( STU \) exactly half of \( PQR \)?

**Tip**: In equilateral triangles, dividing angles through trisectors creates proportional and symmetric segments, often simplifying calculations using trigonometric functions.

The equilateral triangle PQR has side-length 1. The lines PT and PU trisect the angle RPQ, the lines RS and RT trisect the angle QRP, and the lines QS and QU trisect the angle PQR. What is the side-length of the equilateral triangle STU?

Solve a challenging geometry problem involving an equilateral triangle with trisected angles. Learn how to calculate the side-length of a smaller equilateral triangle using trigonometric functions, specifically cosine values. Suitable for advanced high school students in Grades 10-12.

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