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.