Geometry
posted by Rockwell .
What is the equation of the circle touching the lines: 2x +y9=0, 2x +y 1=0 and x +2y +7=0.

Geometry 
Steve
The three lines intersect to form a triangle. Naturally, there is one circle inside the triangle, and three circles outside the triangle, all of which touch the three lines.
The three lines intersect at (2,5),(5,1),(3,5). If we call those vertices A,B,C, then the opposite sides are a,b,c, and we have
Now, the incenter lies at the intersection of the angle bisectors of the vertices. The three lines have slopes 2,2,1/2. Thus, the angle bisectors at A,B,C have slopes undef,1/3,1
So, the center lies on the line x = 2
The other lines are
y+1 = 1/3 (x5)
y+5 = (x+3)
They intersect at (2,0)
The distance from (2,0) to the three original lines is √5, so the circle is
(x2)^2 + y^2 = 5
See the plots at
http://www.wolframalpha.com/input/?i=plot+2x+%2By9%3D0%2C+2x+%2By+1%3D0+%2C+x+%2B2y+%2B7%3D0%2C+%28x2%29^2+%2B+y^2+%3D+5
You can work similar magic if you want to find the excircles. 
Geometry 
MathMate
This problem is very interesting, and as Steve pointed out, there are 4 circles each of which is tangent to all three lines. This prompted me to look for a general solution for all four circles.
If we first examine the condition of tangency of a circle
C: (xa)²+(yb)²=r²
to the line
y=mx+q,
it turns out to be
(bmaq)²=r²(1+m²)
For the circle to be tangent to all three lines,
L1: y=m1x+q1
L2: y=m2x+q2
L3: y=m3x+q3
where
m1=2 q1=9;
m2=2 q2=1;
m3=1/2 q3=7/2;
We can set up the system of equation of three unknowns in a, b and r:
(b+2a9)²=5r²
(b2a1)²=5r²
(ba/2+7/2)²=5r²/4
The solution of which will give the various values of a,b and r.
In particular, we can take squareroot on both sides to give:
b+2a9=sqrt(5)r
b2a1=sqrt(5)r
ba/2+7/2=sqrt(5)r/2
The solution of which is
a=2, b=10, r=3sqrt(5) for the circle below all three lines L1, L2 and L3.
The set
b+2a9=sqrt(5)r
b2a1=sqrt(5)r
ba/2+7/2=sqrt(5)r/2
gives a=13, b=5, r=6sqrt(5) for the circle to the left and above L2 & L3.
The set
b+2a9=sqrt(5)r
b2a1=sqrt(5)r
ba/2+7/2=sqrt(5)r/2
gives a=7, b=5, and r=2sqrt(5) for the circle to the right, and above L1 & L3.
Finally, the set
b+2a9=sqrt(5)r
b2a1=sqrt(5)r
ba/2+7/2=sqrt(5)r/2
gives a=2, b=0 and r=sqrt(5) for the incircle above L3, as obtained by Steve in the previous post. 
Geometry 
MathMate
A plot of the three lines L1, L2 and L3 may be viewed here:
https://imagizer.imageshack.us/v2/500x300q90/674/pUzIJX.png
Respond to this Question
Similar Questions

Analytic Geometry
Find the equation of the circle touching the lines x+2y=4, x+2y=2 and y=2x5 Please I need help, its for my homework 
Analytic Geometry
What is the equation of the circle touching the lines x3y11=0 and 3xy9=0 having its center on the line x+2y+19=0 
Analytic Geometry
What is the equation of the circle touching the lines x3y11=0 and 3xy9=0 having its center on the line x+2y+19=0 
Analytic Geometry
What is the equation of the circle touching the lines x3y11=0 and 3xy9=0 having its center on the line x+2y+19=0 . 
Geometry
Find the equation of a circle centre on the line y=2x+1 touching the y axis and passing through A(4,5) 
math ,geometry
a circle is touching the sides BC of /_\ ABC at P.AB and AC when produced are touching the circle at Q and R respectively, Prove that AQ=1/2 (AB+BC+CA) 
math ,geometry
a circle is touching the sides BC of /_\ ABC at P.AB and AC when produced are touching the circle at Q and R respectively, Prove that AQ=1/2 (AB+BC+CA) 
coordinate geometry
find the equation of circle having centre in 1st quadrant, touching xaxis,having a common tangent y=3^1/2x+4 with the circle x^2+y^2+4x+4y+4=0 such that the distance between two circles along the xaxis is 3 units ? 
analytic geometry
Find the equation of the circle touching the line x + y = 4 at (1,3) and having a radius of sqrt. of 2 solve in two ways. 
Geometry
Which takes up more space: A single circle inscribed in a triangle, touching each side at a single point?