Angle Bisector Plane

The set of points that are equidistant from two intersecting planes consists of their two angle-bisector planes

Suppose the two planes are

\Pi_1:a_1x+b_1y+c_1z+d_1=0

\Pi_2:a_2x+b_2y+c_2z+d_2=0

A point P=(x,y,z) lies on an angle-bisector plane when its distances from the two planes are equal

\frac{|a_1x+b_1y+c_1z+d_1|}{\sqrt{a_1^2+b_1^2+c_1^2}}=\frac{|a_2x+b_2y+c_2z+d_2|}{\sqrt{a_2^2+b_2^2+c_2^2}}

Therefore the two angle-bisector planes are

\frac{a_1x+b_1y+c_1z+d_1}{\sqrt{a_1^2+b_1^2+c_1^2}}=\pm\frac{a_2x+b_2y+c_2z+d_2}{\sqrt{a_2^2+b_2^2+c_2^2}}