a(a+c)=b(b+c)
aa+ac=bb+bc
aa-bb=bc-ac
(a+b)(a-b)=-c(a-b)
Attention! Many people who get the-1 answer here cancel a-b directly, which is wrong. They ignored a problem. When a-b=0, they moved the right one to the left.
(a+b+c)(a-b)=0
a+b+c=0
or
a-b=0
The first case
a+b=-c,k=c/(a+b)=c/(-c)=- 1
The second situation
If you casually bring ab into a formula, you can get a=b=c, k=c/(a+b)=c/(c+c)= 1/2.
So this problem has two values of k,-1 and 1/2.
or
a/b+c=b/a+c=c/a+b=k
a=k(b+c)
b=k(a+c)
c=k(a+b)
amount to
a+b+c=2k(a+b+c)
(a+b+c)(2k- 1)=0
If a+b+c=0
Then b+c =-a.
k=a/(-a)=- 1
If 2k- 1=0
k= 1/2
So k=- 1, k= 1/2.