Equation of the bisector of the angle between two lines containing the given point

Solution 1:

The formula $$ d_1(x,y) = \frac{\lvert a_1x+b_1y+c_1 \rvert}{\sqrt{a_1^2+b_1^2}} $$ gives you the distance from a point $(x,y)$ to the line $L_1$ whose equation is $a_1x+b_1y+c_1 = 0.$ See Distance Between A Point And A Line for a proof of this. For the line $L_2$ given by $a_2x+b_2y+c_2 = 0,$ the distance of a point to the line is given by $$ d_2(x,y) = \frac{\lvert a_2x+b_2y+c_2 \rvert}{\sqrt{a_2^2+b_2^2}}. $$

A point $(x,y)$ on an angle bisector between two lines is equidistant from the two lines, that is, it satisfies the condition $d_1(x,y) = d_2(x,y).$ Writing out the formulas for $d_1$ and $d_2$ in full, $$ \frac{\lvert a_1x+b_1y+c_1 \rvert}{\sqrt{a_1^2+b_1^2}} = \frac{\lvert a_2x+b_2y+c_2 \rvert}{\sqrt{a_2^2+b_2^2}}. $$

Now observe that $\lvert a_1x+b_1y+c_1 \rvert$ will be either $a_1x+b_1y+c_1$ or $-(a_1x+b_1y+c_1),$ whichever of those two expressions is positive. In fact, $a_1x+b_1y+c_1$ will be positive for all points on one side of the line and negative for all points on the other side.

Now if the lines $L_1$ and $L_2$ intersect, they divide the plane into four regions. Label each these regions as $+L_1$ or $-L_1$ depending on whether $a_1x+b_1y+c_1$ is (respectively) positive or negative in that region. Label each region as $+L_2$ or $-L_2$ depending on whether $a_2x+b_2y+c_2$ is (respectively) positive or negative in that region.

One of the angle bisectors of $L_1$ and $L_2$ will go through the regions labeled $+L_1,+L_2$ or $-L_1,-L_2.$ That is, on that line the signs of $a_1x+b_1y+c_1$ and $a_2x+b_2y+c_2$ are either both positive or both negative. Points on this line therefore satisfy the formula $$ \frac{a_1x+b_1y+c_1 }{\sqrt{a_1^2+b_1^2}} = \frac{a_2x+b_2y+c_2 }{\sqrt{a_2^2+b_2^2}}. $$ (For points in the region $-L_1,-L_2,$ this formula gives negative values on both sides, but their absolute values are equal.)

The other angle bisector goes through $+L_1,-L_2$ and $-L_1,+L_2$ and has the formula $$ \frac{a_1x+b_1y+c_1 }{\sqrt{a_1^2+b_1^2}} = - \frac{a_2x+b_2y+c_2 }{\sqrt{a_2^2+b_2^2}}. $$