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. centroid of `DeltaABC`
C. incentre of `DeltaABC`
D. circumcentre of `DletaABC`

Answer

To solve the problem, we need to determine the conditions under which the algebraic sum of the perpendicular distances from three 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 \( (x_i, y_i) \) to a line given by the equation \( ax + by + c = 0 \) is given by:
\[ d_i = \frac{ax_i + by_i + c}{\sqrt{a^2 + b^2}} \]
for \( i = 1, 2, 3 \).

2. **Setting Up the Equation**:
According to the problem, the algebraic sum of the perpendicular distances from points \( A \), \( B \), and \( C \) to the line is zero:
\[ d_1 + d_2 + d_3 = 0 \]
Substituting the distances:
\[ \frac{ax_1 + by_1 + c}{\sqrt{a^2 + b^2}} + \frac{ax_2 + by_2 + c}{\sqrt{a^2 + b^2}} + \frac{ax_3 + by_3 + c}{\sqrt{a^2 + b^2}} = 0 \]

3. **Combining the Terms**:
Since \( \sqrt{a^2 + b^2} \) is a common denominator, we can multiply through by it (assuming it is non-zero):
\[ (ax_1 + by_1 + c) + (ax_2 + by_2 + c) + (ax_3 + by_3 + c) = 0 \]
Simplifying this gives:
\[ a(x_1 + x_2 + x_3) + b(y_1 + y_2 + y_3) + 3c = 0 \]

4. **Finding the Centroid**:
Let \( T = \frac{x_1 + x_2 + x_3}{3} \) and \( K = \frac{y_1 + y_2 + y_3}{3} \). Then, we can rewrite the equation as:
\[ 3c = -a(3T) - b(3K) \]
or:
\[ aT + bK + c = 0 \]
This indicates that the line passes through the point \( (T, K) \), which is the centroid of the triangle formed by points \( A \), \( B \), and \( C \).

5. **Conclusion**:
Therefore, the line passes through the centroid of the triangle formed by the points \( A \), \( B \), and \( C \).

### Final Answer:
The line passes through the centroid of the triangle formed by the points \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \).