Question 752632
triangle determined by lines are x+y=8, 2x+y=14 and 3x+y=22
solving these equations in two groups by elimination 
x+y=8, 2x+y=14  we will get  x=6  and y=2 so vertex coordinates are (6 2)
similarly  2x+y=14 and 3x+y=22  we will get vertex as (8 -2)
           x+y=8,  3x+y=22 we will get vertex  as (7 1)
therefore the circle will pass through vertex of triangle 
circle passing through  (6 2) (8 -2) and (7 1)
we know that general  equation of circle
x^2+y^2+2gx+2fy+c=0

circle passing through(6 2)  will be 
12g+4f+c= -40 (i)

circle passing through   (8 -2)  will be 
16g-4f+c=-68 (ii)
circle passing through   (7 1) we will get 
14g +2f +c =-50 (iii)
by solving equations in two groups
12g+4f+c= -40 (i)
16g-4f+c=-68
 we will get
-4g+8f =28
same way using two equation 
16g-4f+c=-68
14g +2f +c =-50(iv)
 we will get 
2g-6f=-18  (v)
by solving equation (iv) and (v)
we will get 
f=2 and g=-3
by putting value of g and f in equation (iii)
we will get  c=-12
arranging these value of g=-3   f=2  and c=-12 in general equations of circle
equation of circle will be
x^2 +y^2-6x+4y-12 =0
ANSWER equation of circle subscribe the triangle will be
x^2 +y^2-6x+4y-12 =0