SOLUTION: Could you please solve this for me. Your assistance is always honored. * Find the equation of a circle inscribed in atriangle, if the triangle has its sides on the lines; 2x +

2x + y - 9 = 0,   -2x + y - 1= 0, and -x + 2y + 7 = 0.

Let its equation be 
 
The inscribed circle must be such that the perpendicular 
distance from its center (h,k) to each of the three lines 
must all be equal and be equal to the radius r of the 
inscribed circle. 

d = 
 
Use distance from point to line formula:
d = 
 
For the first line
r = 
 
r = 
 
r = 
 
For the second line
 
r = 
 
r = 
 
r = 
 
For the third line
 
r = 
 
r = 
 
r = 
 

 

 
Absolute value equations often have more than one solution, but we are
only interested in a solution that is inside the triangle.  So we can discard
any solution that cannot be inside the triangle.  So we graph the three lines



Setting the first two equal
 

 
 or 
 
 or 
 
 or 
 
 or 

We can disregard k=5 because no point with y-coordinate 5 could be the
y-coordinate of the inscribed circle. However h=2 is a candidate for
the x-coordinate of the radius of the inscribed circle.
 
Setting the first and third equal
 

 
 or 
 
 or 
 
 or 
 
 or 

That's the same. 2 is so far the only candidate for h.  Let's set the second and
third equal:



 or  

 or 
 or 
 or 
 or 

If h=2 then if we substitute in those, we get:

 or 

 or 

 or 

We can disregard -10 so the center of the inscribed circle must be

(h,k) = (2.0)

So we find the radius by substituting in any one of the three equations
for the radius, say, the first one

r = 
r = 
r = 
r = 

r = 

rationalize the denominator

r = 

That makes



So the equation is



{{x-2)^2+(y-0)^2=5}}}

{{x-2)^2+y^2=5}}}



Edwin