is the point one third of the distance from (x1,y1) to (x2,y2) sometimes always or never the point (x1+x2 over 3, y1+y2 over 3) 

To determine whether the point one-third of the distance from \((x_1, y_1)\) to \((x_2, y_2)\) is sometimes, always, or never \(\left(\frac{x_1 + x_2}{3}, \frac{y_1 + y_2}{3}\)\), we need to consider how points divide a line segment in a particular ratio.

### Step 1: Calculate the point that is one-third of the distance from \((x_1, y_1)\) to \((x_2, y_2)\).
The point one-third of the way from \((x_1, y_1)\) to \((x_2, y_2)\) divides the line segment in a ratio of \(1:2\). The formula for the coordinates of a point dividing a line segment in the ratio \(m:n\) is given by:

\[
\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\]

For \(m = 1\) and \(n = 2\), the coordinates of the point are:

\[
\left( \frac{1 \cdot x_2 + 2 \cdot x_1}{1+2}, \frac{1 \cdot y_2 + 2 \cdot y_1}{1+2} \right) = \left( \frac{x_2 + 2x_1}{3}, \frac{y_2 + 2y_1}{3} \right)
\]

### Step 2: Compare this with the point \(\left(\frac{x_1 + x_2}{3}, \frac{y_1 + y_2}{3}\right)\).
The given point is \(\left(\frac{x_1 + x_2}{3}, \frac{y_1 + y_2}{3}\)\). This point represents the coordinates obtained by averaging \((x_1, y_1)\) and \((x_2, y_2)\) and dividing by 3.

### Step 3: Analyze the conditions.
- **Always**: This would mean that the two points are always equal, which is not the case, as shown by the different formulas.
- **Never**: The two points might coincide under certain conditions (e.g., when \((x_1, y_1) = (x_2, y_2)\)), but generally, they don't coincide.
- **Sometimes**: The two points coincide under specific conditions, such as when \(x_1 = x_2\) and \(y_1 = y_2\), or when the line segment happens to be horizontal or vertical with appropriate coordinates.

### Conclusion:
The point one-third of the distance from \((x_1, y_1)\) to \((x_2, y_2)\) is **sometimes** the point \(\left(\frac{x_1 + x_2}{3}, \frac{y_1 + y_2}{3}\right)\), but only under special conditions.

Would you like more details or have any questions?

Here are 5 related questions you might explore:
1. How do you find the midpoint of a line segment?
2. What is the formula for dividing a line segment into a given ratio?
3. How would you calculate the centroid of a triangle?
4. What happens to the coordinates if you divide a line segment into a different ratio, like 2:3?
5. How does the distance formula relate to finding points along a line?

**Tip:** Understanding the concept of dividing a line segment into specific ratios is key to solving problems related to points on a line.

Explore whether the point one-third of the distance from (x1, y1) to (x2, y2) is sometimes, always, or never the point (x1 + x2)/3, (y1 + y2)/3. This detailed mathematical analysis covers the formula for dividing a line segment, with practical examples and explanations. Suitable for high school students and above.

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