12.0 Excentre of a triangle

The point of intersection of the external bisectors of the angles of a triangle is called the excentre.

If $A({x_1},{y_1})$, $B({x_2},{y_2})$ and $C({x_3},{y_3})$ are the vertices of a $\Delta ABC$, whose sides $BC$, $CA$ and $AB$ are of lengths $a$, $b$ and $c$ respectively, then co-ordinates of incentre of $\Delta ABC$ is $$\begin{equation} \begin{aligned} {I_1} = (\frac{{ - a{x_1} + b{x_2} + c{x_3}}}{{ - a + b + c}},\frac{{ - a{y_1} + b{y_2} + c{y_3}}}{{ - a + b + c}}) \\ {I_2} = (\frac{{a{x_1} - b{x_2} + c{x_3}}}{{a - b + c}},\frac{{a{y_1} - b{y_2} + c{y_3}}}{{a - b + c}}) \\ {I_3} = (\frac{{a{x_1} + b{x_2} - c{x_3}}}{{a + b - c}},\frac{{a{y_1} + b{y_2} - c{y_3}}}{{a + b - c}}) \\\end{aligned} \end{equation} $$