Co-ordinate Geometry(Incenter)

Question

Is there any way of finding the incenter of a triangle, when the equation of all three lines of the triangle are given, without finding out the vertices?

Answer 1

Solving $$ \frac{a_{1} X+b_{1}Y+c_{1}}{\sqrt{a_{1}^{2}+b_{1}^{2}}} =\pm \frac{a_{2} X+b_{2}Y+c_{2}}{\sqrt{a_{2}^{2}+b_{2}^{2}}} =\pm \frac{a_{3} X+b_{3}Y+c_{3}}{\sqrt{a_{3}^{2}+b_{3}^{2}}} $$

gives one in-centre and three ex-centres.