Bisector of the Angles Between Two Lines

Introduction

Bisector of the angles between two lines (or Angle bisector of two lines) are the lines which bisects the angle between the two given lines. These Angle bisector of two lines are the locus of a point which is equidistant from the two lines. In other words, an angle bisector has equal perpendicular distance from the two lines.

Equation of Angle Bisector

Let us consider a pair of straight lines given by
\(l_1 : a_1x + b_1y + c_1= 0\) and \(l_2 : a_2x + b_2y + c_2= 0\)
Also let, P(x, y) be a point lies on the angle bisectors, then length of perpendicular from the point P to both the lines \(l_1,l_2\) are equal.
Thus
\( \frac{a_1x + b_1y + c_1}{\sqrt{a_1^2+b_1^2}}=\pm \frac{a_2x + b_2y + c_2}{\sqrt{a_2^2+b_2^2}}\)(1)
Solving (1), we will get two bisectors \(b_1,b_2\) as required.