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This problem is about finding various characteristics of triangle \(ABC\), given the vertices \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \). Here’s a breakdown of what each part requires:

1. **Equation of side \( AB \)**:
   - Use the two-point form of the line equation for points \( A(x_1, y_1) \) and \( B(x_2, y_2) \).

2. **Equation of altitude \( CH \)**:
   - The altitude \( CH \) is perpendicular to \( AB \) and passes through point \( C(x_3, y_3) \). First, determine the slope of \( AB \), then use the negative reciprocal for the slope of \( CH \) to form its equation.

3. **Equation of median \( AM \)**:
   - The median \( AM \) connects vertex \( A(x_1, y_1) \) to the midpoint \( M \) of side \( BC \). Calculate \( M \)'s coordinates using the midpoint formula: \( M\left(\frac{x_2 + x_3}{2}, \frac{y_2 + y_3}{2}\right) \), then form the line equation from \( A \) to \( M \).

4. **Point of intersection \( N \) of median \( AM \) and altitude \( CH \)**:
   - Solve the equations of \( AM \) and \( CH \) simultaneously to find the coordinates of point \( N \).

5. **Equation of line through \( C \) parallel to \( AB \)**:
   - Use the slope of \( AB \) for this line, as parallel lines have the same slope. The line passes through \( C(x_3, y_3) \).

6. **Distance from point \( C \) to line \( AB \)**:
   - Apply the point-to-line distance formula to find the perpendicular distance from \( C(x_3, y_3) \) to the line \( AB \).

Would you like detailed solutions for each part, or do you need help with a specific item?

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Here are some follow-up questions for deeper understanding:

1. How do we derive the equation of a line from two points?
2. Why is the altitude perpendicular to the opposite side, and how does this affect the equation?
3. What is the significance of the median and its intersection properties in a triangle?
4. How do we find the intersection of two lines mathematically?
5. What role does the slope play in determining parallel and perpendicular lines?

**Tip**: To ensure accuracy, double-check the calculations for slopes and line equations, especially when working with complex coordinates.

Given the vertices of triangle ABC: A(x1, y1), B(x2, y2), C(x3, y3), find:
   a) Equation of side AB;
   b) Equation of altitude CH;
   c) Equation of median AM;
   d) Point N, intersection of median AM and altitude CH;
   e) Equation of line passing through vertex C parallel to side AB;
   f) Distance from point C to line AB.

Explore step-by-step solutions for finding the equations of side AB, altitude CH, median AM, and the line passing through vertex C parallel to side AB in a triangle. Learn how to determine intersection points and calculate the distance from a point to a line, covering core concepts in analytic geometry. Suitable for high school students (Grades 10-12).

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