If you choose appropriate directions, the angle between a vector on ax+by+c=0 (you know what this looks like) and another vector on the bisector should be 45 degrees. Note that the length of the vectors doesn’t matter here. Using dot product you can solve for the vector on the bisector, which determines the direction of the bisector. Find a point on the bisector and use it to write a parametric equation for the bisector. Transform that to normal equation.