A triangle has one red side, one blue side, and one green side. Jane draws three red lines parallel to the red side, two blue lines parallel to the blue side, and one green line parallel to the green side. (Every pair of lines that are different colors intersect inside the triangle. And no three lines intersect at the same point.) In how many parts is the triangle split by these six lines?

To solve this problem, let’s go through the steps systematically.

1. **Understanding the Line Intersections:**  
   - Jane draws three red lines parallel to the red side of the triangle.
   - She draws two blue lines parallel to the blue side.
   - She draws one green line parallel to the green side.

2. **Key Observations:**
   - Every pair of lines with different colors will intersect within the triangle.
   - No three lines intersect at a single point.

3. **Calculating the Number of Parts Formed:**
   The total number of regions created by multiple lines intersecting within a triangular region can be calculated using a general formula for regions created by multiple sets of intersecting parallel lines:
   \[
   R = (r+1)(b+1)(g+1)
   \]
   where:
   - \( r \) is the number of lines parallel to the red side (here, \( r = 3 \)),
   - \( b \) is the number of lines parallel to the blue side (here, \( b = 2 \)),
   - \( g \) is the number of lines parallel to the green side (here, \( g = 1 \)).

4. **Applying the Formula:**
   Substituting the values, we get:
   \[
   R = (3+1)(2+1)(1+1) = 4 \times 3 \times 2 = 24
   \]

So, the triangle is divided into **24 parts** by these six lines.

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Would you like a deeper explanation of this process or details about how the formula is derived?

### Related Questions
1. What if Jane had drawn one additional blue line?
2. How does this formula generalize if we have more than three colors?
3. How would the solution change if some lines didn’t intersect within the triangle?
4. Is there a way to visualize these intersecting lines to understand the parts they form?
5. How would you approach this if no formula were available?

#### Tip:
For problems involving intersecting lines, visualize or sketch small cases to observe patterns before applying a general formula.

Find out how a triangle is divided into parts by parallel lines of different colors intersecting inside the triangle. This geometry problem explores intersecting lines and regions formed, using combinatorial calculations with a detailed solution approach. Suitable for high school students in Grades 10-12.

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