Question 1208632: find the equation of the circle inscribed in a triangle with the sides on the lines x-3y=-5, 3x+y=1 and 3x-y=-11
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Let's label the lines as  ,  , and 
Furthermore, let's say the following
point A = intersection of &
point B = intersection of &
point C = intersection of &
L1 in red, L2 in green, L3 in blue
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The formula for the shortest distance between the point (p,q) and the line ax+by+c = 0 is:
Note: if (p,q) is on the line ax+by+c=0, then the expression ap+bq+c evaluates to 0, which leads to d = 0.
This page shows the derivation of the formula.
Here's a page showing a proof
The angle bisector is a locus of points that are equidistant from each line.
An example would be to think of the xy axis. An angle bisector of the x axis and y axis would be the line y = x. Any point on y = x is equally distant to either axis.
A point like (5,5) is 5 units away from either axis.
This equation
will allow us to find the equation of the angle bisector between the lines ax+by+c = 0 and ex+fy+g = 0. Note that the lines cannot be parallel.
A point (p,q) satisfies this equation if and only if it is equally distant between the lines. We can think of it like a midpoint.
Notice the left hand side and right hand side match the template of the point-to-line distance formula mentioned.
Rewrite x-3y=-5, 3x+y=1 as x-3y+5=0, 3x+y-1=0 so we can use them in the formula below.
 or 
Solving for y gets us 
This is the bisector of angle CAB.
Part of this bisector is inside triangle ABC. If we solved the other equation for y, then we'd get an angle bisector entirely outside of triangle ABC, which means we can ignore it.
Follow similar steps to find the bisector of angle CBA is the equation 
This is a vertical line through point B.
The intersection of the angle bisectors will give us the center of the inscribed circle.
This is known as the incenter.
Solve this system  to get (x,y) = (-5/3, 7/3) which is the incenter.
Call this point D.
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Once again recall that this formula
gives the distance between the line ax+by+c = 0 and the point (p,q)
We want to find the shortest distance from point D(-5/3, 7/3) to any of the three given lines. I'll use the line 3x+y-1 = 0.
Determining this distance will give us the radius of the inscribed circle (known as the inradius).
This is inradius.
So we have  which has both sides square to 
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Recap
We found that the incenter is at (-5/3, 7/3). These are the h,k values.
The r^2 value is 121/90
Therefore we go from  to the final answer 
I used GeoGebra to verify the answer.
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