The algebraic sum of the perpendicular distances from $A\left(x_1,y_1\right),B\left(x_2,y_2\right)andC\left(x_3,y_3\right)$ to a variable line is zero, then the line passes through

Let Equation of line ax + by + c = 0—(1)

$\begin{array}{l}{p}_1=\perp rdis\tan cefromA\left(x_1,y_1\right)to\left(1\right)\Rightarrow \frac{\left|a{x}_1+b{y}_1+c\right|}{\sqrt{a^2+b^2}}\\ p_2=\perp rdis\tan cefromB\left(x_2,y_2\right)to\left(1\right)\Rightarrow \frac{\left|a{x}_2+b{y}_2+c\right|}{\sqrt{a^2+b^2}}\\ p_3=\perp rdis\tan cefromC\left(x_3,y_3\right)to\left(1\right)\Rightarrow \frac{\left|a{x}_3+b{y}_3+c\right|}{\sqrt{a^2+b^2}}\end{array}$

Given p₁+p₂+p₃=0

$\begin{array}{l}\Rightarrow \frac{\left|a{x}_1+b{y}_1+c\right|}{\sqrt{a^2+b^2}}+\frac{\left|a{x}_2+b{y}_2+c\right|}{\sqrt{a^2+b^2}}+\frac{\left|a{x}_3+b{y}_3+c\right|}{\sqrt{a^2+b^2}}=0\\ \Rightarrow a\left(x_1+x_2+x_3\right)+b\left(y_1+y_2+y_3\right)+3c=0\\ \Rightarrow a\left(\frac{x_1+x_2+x_3}{3}\right)+b\left(\frac{y_1+y_2+y_3}{3}\right)+c=0\\ Thelinepas\sin gthrough\left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\right)=G\end{array}$