The algebraic sum of the perpendicular distance from `A(x_(1),y_(1)),B(x_(2),y_(2))` and `C(x_(3),y_(3))` to a variable line is zero, then the line passes through

Options

A. the orthocentre of `DeltaABC`
B. the centroid of `DeltaABC`
C. the circumcentre of `DeltaABC`
D. None of these

Answer

To solve the problem, we need to determine the conditions under which the algebraic sum of the perpendicular distances from points \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \) to a variable line is zero.

### Step-by-Step Solution:

1. **Understanding the Perpendicular Distance**:
The perpendicular distance \( d \) from a point \( (x_0, y_0) \) to the line \( Ax + By + C = 0 \) is given by the formula:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]

2. **Setting Up the Equation**:
We need to find the sum of the perpendicular distances from points \( A \), \( B \), and \( C \) to the line, which should equal zero:
\[ d_A + d_B + d_C = 0 \]
Substituting the formula for distance, we have:
\[ \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} + \frac{|Ax_2 + By_2 + C|}{\sqrt{A^2 + B^2}} + \frac{|Ax_3 + By_3 + C|}{\sqrt{A^2 + B^2}} = 0 \]

3. **Simplifying the Expression**:
Since \( \sqrt{A^2 + B^2} \) is a positive quantity (it cannot be zero), we can multiply through by \( \sqrt{A^2 + B^2} \) to eliminate the denominator:
\[ |Ax_1 + By_1 + C| + |Ax_2 + By_2 + C| + |Ax_3 + By_3 + C| = 0 \]

4. **Analyzing the Absolute Values**:
The only way for the sum of absolute values to equal zero is if each individual term is zero:
\[ Ax_1 + By_1 + C = 0 \]
\[ Ax_2 + By_2 + C = 0 \]
\[ Ax_3 + By_3 + C = 0 \]

5. **Conclusion**:
If the algebraic sum of the perpendicular distances is zero, then the line must pass through all three points \( A \), \( B \), and \( C \). However, if we consider the centroid of triangle \( ABC \), which is given by:
\[ \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]
The line will pass through the centroid of triangle \( ABC \).

### Final Answer:
The line passes through the centroid of triangle \( ABC \).