You can first divide the formula in the topic into three parts, as shown below:

a+b=ck①，

b+c=ak②，

c+a=bk③

Then add these three formulas:

Add the left to the left and the right to the right to get the following formula:

a+b+b+c+c+a=bk+ak+ck

Integration, you can get:

2(a+b+c)=k(a+b+c)

So to simplify, there are two situations:

(1) When a+b+c=0 and a+b=-c, substitute a+b=-c into the first formula a+b=ck, and you can get:

-c=ck, so-1=k, so k=- 1.

(You can also convert a+b+c=0 into b+c=-a or a+c=-b, and then substitute it into the second formula or the first formula to get k=- 1).

② When a+b+c≠0, the left and right sides of the equation are divided by the same number that is not 0 at the same time, and the value of the equation remains unchanged.

So 2=k, so k=2.

To sum up, k has two values: k=2 or-1.