Question 903672
<pre>
 x + 2y - 3 = 0 
2x +  y - 4 = 0

These are the black lines in the graph below:

{{{drawing(200,200,-5,8,-5,8,graph(200,200,-5,8,-5,8,(x-1)*sqrt(sin(9x))/sqrt(sin(9x))),line(-9,6,9,-3),
green(line(5.6,4.6,3.24,-.12),line(5.6,4.6,.88,2.24)),locate(5.6,4.7,"(x,y)"),
line(-9,22,8,-12) )}}}

Let the variable point (x,y) be any point on the bisector of the
obtuse angle between the two black lines.

Then the perpendicular distances from (x,y) to each of 
those lines (the lengths of the two green line segments) are equal. 

The formula for the perpendicular distance from the point (x<sub>1</sub>,y<sub>1</sub>)
to the line Ax+y+C=0 is:

d = {{{abs(Ax[1]+By[1]+C)/sqrt(A^2+B^2)}}}

So we substitute (x<sub>1</sub>,y<sub>1</sub>) = (x,y) 
for the first line:

x + 2y - 3 = 0,  A=1, B=2, C=-3

d = {{{abs(1x+2y-3)/sqrt(1^2+2^2)}}}
 
d = {{{abs(x+2y-3)/sqrt(1+4)}}}

d = {{{abs(x+2y-3)/sqrt(5)}}}

And also for the second line:

2x +  y - 4 = 0,  A=2, B=1, C=-4

d = {{{abs(2x+1y-4)/sqrt(2^2+1^2)}}}

d = {{{abs(2x+y-4)/sqrt(4+1)}}}

d = {{{abs(2x+y-4)/sqrt(5)}}}

We set those distances equal:

{{{abs(x+2y-3)/sqrt(5)}}}{{{""=""}}}{{{abs(2x+y-4)/sqrt(5)}}}

Multiply both sides by {{{sqrt(5)}}}

{{{abs(x+2y-3)}}}{{{""=""}}}{{{abs(2x+y-4)}}}

This breaks into two equations.  One is the angle bisector of the
acute angle and the other is the equation of the bisector of the
obtuse angle (the one we want).

{{{x+2y-3}}}{{{""=""}}}{{{2x+y-4}}}  and  {{{x+2y-3}}}{{{""=""}}}{{{-(2x+y-4)}}}
{{{x+2y-3}}}{{{""=""}}}{{{2x+y-4}}}  and  {{{x+2y-3}}}{{{""=""}}}{{{-(2x+y-4)}}}
{{{-x+y+1=0}}}                      and  {{{x+2y-3}}}{{{""=""}}}{{{-2x-y+4}}} 
                                     and  {{{3x+3y-7=0}}}

We can see by the graph that the dotted line has a positive slope, and
that the line bisecting the acute angle between the two black lines has
a negative slope.

We put the two lines in slope intercept form

{{{-x+y+1=0}}}    {{{3x+3y-7=0}}}
   {{{y=x-1}}}       {{{3y=-3x+7}}}
                     {{{y=-x+7/3}}}

And we see that the first one has a positive slope, so the answer
is 

{{{y=x-1}}}

The other line {{{y=-x+7/3}}} is the purple dotted line below.
It has a negative slope, which as you can see, bisects the acute 
angle between the two black lines:

{{{drawing(200,200,-5,8,-5,8,graph(200,200,-5,8,-5,8,(x-1)*sqrt(sin(9x))/sqrt(sin(9x))),line(-9,6,9,-3),
green(line(5.6,4.6,3.24,-.12),line(5.6,4.6,.88,2.24)),locate(5.6,4.7,"(x,y)"),
line(-9,22,8,-12),graph(200,200,-5,8,-5,8,20,15,(-x+7/3)*sqrt(sin(9x))/sqrt(sin(9x)))


 )}}}

But the answer that we want is the red dotted line, which 
has equation:

{{{y=x-1}}}

Edwin</pre>