Triangle/Angle bisectors/Intersecting point/Fact

The three angle bisectors in a nondegenerate triangle

If the vertices are given by ${\displaystyle {}A={\begin{pmatrix}a_{1}\\a_{2}\end{pmatrix}},B={\begin{pmatrix}b_{1}\\b_{2}\end{pmatrix}},C={\begin{pmatrix}c_{1}\\c_{2}\end{pmatrix}}}$, and if the side lengths are denoted by ${\displaystyle {}a=d(B,C),b=d(A,C),c=d(A,B)}$, then this intersecting point has the coordinates

${\displaystyle {\frac {1}{a+b+c}}{\begin{pmatrix}aa_{1}+bb_{1}+cc_{1}\\aa_{2}+bb_{2}+cc_{2}\end{pmatrix}}.}$