If two lines are perpendicular to each other and a third line bisects the right angle, then what would be the equation of that bisector?
I mean, I know the equations for acute and obtuse angle as well as the methods to differentiate them. Also, I can find the equation for that right angle bisector using slope. But I want to know if I can find this out from the equations of angle bisector, or if there is any rule of signs of the constants/coefficients of x and y.
Let the point of intersection of the two perpendicular lines have the coordinates $\,(x_0,y_0).\,$ Let the slope of one of the two perpendicular lines be denoted by $\,s=\tan(\theta).\,$ Then the slope of the other is $\,-1/s.\,$ The addition (subtraction) theorem for tangent (slope) is $$ \tan(a\pm b) = \frac{\tan(a)\pm\tan(b)}{1\mp\tan(a)\tan(b)}. $$ Because the tangent of $\,\frac12 90^\circ=45^\circ\,$ is $\,1,\,$ then the tangent of the bisector is $$ \tan(\theta\pm45^\circ) = \frac{s\pm1}{1\mp s}.$$ The $\,\pm\,$ is because there are two perpendicular bisectors. The equation of a line with slope $\,s\,$ passing through $\,(x_0,y_0)\,$ is $\, (y-y_0) = s(x-x_0).\,$ The other lines have similar equations but with different slopes.